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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is_preconnected.Ioi_cInf_subset {s : set α} (hs : is_preconnected s) (hb : bdd_below s)
(ha : ¬bdd_above s) :
Ioi (Inf s) ⊆ s | begin
have sne : s.nonempty := @nonempty_of_not_bdd_above α _ s ⟨Inf ∅⟩ ha,
intros x hx,
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := (is_glb_lt_iff (is_glb_cInf sne hb)).1 hx,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end | lemma | is_preconnected.Ioi_cInf_subset | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"is_glb_cInf",
"is_glb_lt_iff",
"is_preconnected",
"nonempty_of_not_bdd_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.Iio_cSup_subset {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : bdd_above s) :
Iio (Sup s) ⊆ s | @is_preconnected.Ioi_cInf_subset αᵒᵈ _ _ _ s hs ha hb | lemma | is_preconnected.Iio_cSup_subset | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"is_preconnected",
"is_preconnected.Ioi_cInf_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.mem_intervals {s : set α} (hs : is_preconnected s) :
s ∈ ({Icc (Inf s) (Sup s), Ico (Inf s) (Sup s), Ioc (Inf s) (Sup s), Ioo (Inf s) (Sup s),
Ici (Inf s), Ioi (Inf s), Iic (Sup s), Iio (Sup s), univ, ∅} : set (set α)) | begin
rcases s.eq_empty_or_nonempty with rfl|hne,
{ apply_rules [or.inr, mem_singleton] },
have hs' : is_connected s := ⟨hne, hs⟩,
by_cases hb : bdd_below s; by_cases ha : bdd_above s,
{ rcases mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_cInf_cSup_subset hb ha)
(subset_Icc_cInf_cSup hb ha) with hs|... | lemma | is_preconnected.mem_intervals | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"cInf_le",
"is_connected",
"is_preconnected",
"le_cSup",
"subset_Icc_cInf_cSup"
] | A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordererd. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_of_is_preconnected_subset_of_ordered :
{s : set α | is_preconnected s} ⊆
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) | begin
intros s hs,
rcases hs.mem_intervals with hs|hs|hs|hs|hs|hs|hs|hs|hs|hs,
{ exact (or.inl $ or.inl $ or.inl $ or.inl ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ o... | lemma | set_of_is_preconnected_subset_of_ordered | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though
one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of possible cases to improve
readability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.mem_of_ge_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).nonempty) :
b ∈ s | begin
let S := s ∩ Icc a b,
replace ha : a ∈ S, from ⟨ha, left_mem_Icc.2 hab⟩,
have Sbd : bdd_above S, from ⟨b, λ z hz, hz.2.2⟩,
let c := Sup (s ∩ Icc a b),
have c_mem : c ∈ S, from hs.cSup_mem ⟨_, ha⟩ Sbd,
have c_le : c ≤ b, from cSup_le ⟨_, ha⟩ (λ x hx, hx.2.2),
cases eq_or_lt_of_le c_le with hc hc, fro... | lemma | is_closed.mem_of_ge_of_forall_exists_gt | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"cSup_le",
"eq_or_lt_of_le",
"is_closed",
"le_cSup",
"not_lt_of_le"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.Icc_subset_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).nonempty) :
Icc a b ⊆ s | begin
assume y hy,
have : is_closed (s ∩ Icc a y),
{ suffices : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y,
{ rw this, exact is_closed.inter hs is_closed_Icc },
rw [inter_assoc],
congr,
exact (inter_eq_self_of_subset_right $ Icc_subset_Icc_right hy.2).symm },
exact is_closed.mem_of_ge_of_forall_exists_... | lemma | is_closed.Icc_subset_of_forall_exists_gt | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_closed",
"is_closed.inter",
"is_closed.mem_of_ge_of_forall_exists_gt",
"is_closed_Icc"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.Icc_subset_of_forall_mem_nhds_within {a b : α} {s : set α}
(hs : is_closed (s ∩ Icc a b)) (ha : a ∈ s)
(hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) :
Icc a b ⊆ s | begin
apply hs.Icc_subset_of_forall_exists_gt ha,
rintros x ⟨hxs, hxab⟩ y hyxb,
have : s ∩ Ioc x y ∈ 𝓝[>] x,
from inter_mem (hgt x ⟨hxs, hxab⟩) (Ioc_mem_nhds_within_Ioi ⟨le_rfl, hyxb⟩),
exact (nhds_within_Ioi_self_ne_bot' ⟨b, hxab.2⟩).nonempty_of_mem this
end | lemma | is_closed.Icc_subset_of_forall_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"Ioc_mem_nhds_within_Ioi",
"is_closed",
"nhds_within_Ioi_self_ne_bot'"
] | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_Icc_aux (x y : α) (s t : set α) (hxy : x ≤ y)
(hs : is_closed s) (ht : is_closed t) (hab : Icc a b ⊆ s ∪ t)
(hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) : (Icc a b ∩ (s ∩ t)).nonempty | begin
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2,
by_contradiction hst,
suffices : Icc x y ⊆ s,
from hst ⟨y, xyab $ right_mem_Icc.2 hxy, this $ right_mem_Icc.2 hxy, hy.2⟩,
apply (is_closed.inter hs is_closed_Icc).Icc_subset_of_forall_mem_nhds_within hx.2,
rintros z ⟨zs, hz⟩,
have zt :... | lemma | is_preconnected_Icc_aux | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"by_contradiction",
"is_closed",
"is_closed.inter",
"is_closed_Icc",
"nhds_within_Ioc_eq_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Icc : is_preconnected (Icc a b) | is_preconnected_closed_iff.2
begin
rintros s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩,
-- This used to use `wlog`, but it was causing timeouts.
cases le_total x y,
{ exact is_preconnected_Icc_aux x y s t h hs ht hab hx hy, },
{ rw inter_comm s t,
rw union_comm s t at hab,
exact is_preconnected_Icc_aux y x t s h ht... | lemma | is_preconnected_Icc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected",
"is_preconnected_Icc_aux"
] | A closed interval in a densely ordered conditionally complete linear order is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_uIcc : is_preconnected (uIcc a b) | is_preconnected_Icc | lemma | is_preconnected_uIcc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected",
"is_preconnected_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.is_preconnected {s : set α} (h : s.ord_connected) :
is_preconnected s | is_preconnected_of_forall_pair $ λ x hx y hy, ⟨uIcc x y, h.uIcc_subset hx hy,
left_mem_uIcc, right_mem_uIcc, is_preconnected_uIcc⟩ | lemma | set.ord_connected.is_preconnected | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected",
"is_preconnected_of_forall_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_iff_ord_connected {s : set α} :
is_preconnected s ↔ ord_connected s | ⟨is_preconnected.ord_connected, set.ord_connected.is_preconnected⟩ | lemma | is_preconnected_iff_ord_connected | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Ici : is_preconnected (Ici a) | ord_connected_Ici.is_preconnected | lemma | is_preconnected_Ici | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Iic : is_preconnected (Iic a) | ord_connected_Iic.is_preconnected | lemma | is_preconnected_Iic | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Iio : is_preconnected (Iio a) | ord_connected_Iio.is_preconnected | lemma | is_preconnected_Iio | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Ioi : is_preconnected (Ioi a) | ord_connected_Ioi.is_preconnected | lemma | is_preconnected_Ioi | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Ioo : is_preconnected (Ioo a b) | ord_connected_Ioo.is_preconnected | lemma | is_preconnected_Ioo | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Ioc : is_preconnected (Ioc a b) | ord_connected_Ioc.is_preconnected | lemma | is_preconnected_Ioc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_Ico : is_preconnected (Ico a b) | ord_connected_Ico.is_preconnected | lemma | is_preconnected_Ico | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Ici : is_connected (Ici a) | ⟨nonempty_Ici, is_preconnected_Ici⟩ | lemma | is_connected_Ici | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Iic : is_connected (Iic a) | ⟨nonempty_Iic, is_preconnected_Iic⟩ | lemma | is_connected_Iic | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Ioi [no_max_order α] : is_connected (Ioi a) | ⟨nonempty_Ioi, is_preconnected_Ioi⟩ | lemma | is_connected_Ioi | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Iio [no_min_order α] : is_connected (Iio a) | ⟨nonempty_Iio, is_preconnected_Iio⟩ | lemma | is_connected_Iio | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Icc (h : a ≤ b) : is_connected (Icc a b) | ⟨nonempty_Icc.2 h, is_preconnected_Icc⟩ | lemma | is_connected_Icc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Ioo (h : a < b) : is_connected (Ioo a b) | ⟨nonempty_Ioo.2 h, is_preconnected_Ioo⟩ | lemma | is_connected_Ioo | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Ioc (h : a < b) : is_connected (Ioc a b) | ⟨nonempty_Ioc.2 h, is_preconnected_Ioc⟩ | lemma | is_connected_Ioc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_Ico (h : a < b) : is_connected (Ico a b) | ⟨nonempty_Ico.2 h, is_preconnected_Ico⟩ | lemma | is_connected_Ico | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_connected_space : preconnected_space α | ⟨ord_connected_univ.is_preconnected⟩ | instance | ordered_connected_space | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_is_preconnected_eq_of_ordered :
{s : set α | is_preconnected s} =
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) | begin
refine subset.antisymm set_of_is_preconnected_subset_of_ordered _,
simp only [subset_def, -mem_range, forall_range_iff, uncurry, or_imp_distrib, forall_and_distrib,
mem_union, mem_set_of_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
is_preconnected_Icc, is_preconnected_Ico, i... | lemma | set_of_is_preconnected_eq_of_ordered | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"forall_and_distrib",
"forall_eq",
"forall_true_iff",
"is_preconnected",
"is_preconnected_Icc",
"is_preconnected_Ici",
"is_preconnected_Ico",
"is_preconnected_Iic",
"is_preconnected_Iio",
"is_preconnected_Ioc",
"is_preconnected_Ioi",
"is_preconnected_Ioo",
"is_preconnected_empty",
"or_imp_... | In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of
possible cases to improve readability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f a) (f b) ⊆ f '' (Icc a b) | is_preconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf | lemma | intermediate_value_Icc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≤ t ≤ f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f b) (f a) ⊆ f '' (Icc a b) | is_preconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf | lemma | intermediate_value_Icc' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≥ t ≥ f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intermediate_value_uIcc {a b : α} {f : α → δ} (hf : continuous_on f (uIcc a b)) :
uIcc (f a) (f b) ⊆ f '' uIcc a b | by cases le_total (f a) (f b); simp [*, is_preconnected_uIcc.intermediate_value] | lemma | intermediate_value_uIcc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ico (f a) (f b) ⊆ f '' (Ico a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.2 (not_lt_of_le (he ▸ h.1)))
(λ hlt, @is_preconnected.intermediate_value_Ico _ _ _ _ _ _ _ (is_preconnected_Ico)
_ _ ⟨refl a, hlt⟩ (right_nhds_within_Ico_ne_bot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self)
_ ((hf.continuous_within_at ⟨hab, refl b⟩).mono Ico_subs... | lemma | intermediate_value_Ico | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ico",
"is_preconnected_Ico",
"not_lt_of_le",
"right_nhds_within_Ico_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ioc (f b) (f a) ⊆ f '' (Ico a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.1 (not_lt_of_le (he ▸ h.2)))
(λ hlt, @is_preconnected.intermediate_value_Ioc _ _ _ _ _ _ _ (is_preconnected_Ico)
_ _ ⟨refl a, hlt⟩ (right_nhds_within_Ico_ne_bot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self)
_ ((hf.continuous_within_at ⟨hab, refl b⟩).mono Ico_subs... | lemma | intermediate_value_Ico' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ioc",
"is_preconnected_Ico",
"not_lt_of_le",
"right_nhds_within_Ico_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ioc (f a) (f b) ⊆ f '' (Ioc a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.2 (not_le_of_lt (he ▸ h.1)))
(λ hlt, @is_preconnected.intermediate_value_Ioc _ _ _ _ _ _ _ (is_preconnected_Ioc)
_ _ ⟨hlt, refl b⟩ (left_nhds_within_Ioc_ne_bot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self)
_ ((hf.continuous_within_at ⟨refl a, hab⟩).mono Ioc_subse... | lemma | intermediate_value_Ioc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ioc",
"is_preconnected_Ioc",
"left_nhds_within_Ioc_ne_bot",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ico (f b) (f a) ⊆ f '' (Ioc a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.1 (not_le_of_lt (he ▸ h.2)))
(λ hlt, @is_preconnected.intermediate_value_Ico _ _ _ _ _ _ _ (is_preconnected_Ioc)
_ _ ⟨hlt, refl b⟩ (left_nhds_within_Ioc_ne_bot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self)
_ ((hf.continuous_within_at ⟨refl a, hab⟩).mono Ioc_subse... | lemma | intermediate_value_Ioc' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ico",
"is_preconnected_Ioc",
"left_nhds_within_Ioc_ne_bot",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ioo (f a) (f b) ⊆ f '' (Ioo a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.2 (not_lt_of_lt (he ▸ h.1)))
(λ hlt, @is_preconnected.intermediate_value_Ioo _ _ _ _ _ _ _ (is_preconnected_Ioo)
_ _ (left_nhds_within_Ioo_ne_bot hlt) (right_nhds_within_Ioo_ne_bot hlt)
inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self)
_ _ ((hf.continuous_w... | lemma | intermediate_value_Ioo | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ioo",
"is_preconnected_Ioo",
"left_nhds_within_Ioo_ne_bot",
"right_nhds_within_Ioo_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Ioo (f b) (f a) ⊆ f '' (Ioo a b) | or.elim (eq_or_lt_of_le hab) (λ he y h, absurd h.1 (not_lt_of_lt (he ▸ h.2)))
(λ hlt, @is_preconnected.intermediate_value_Ioo _ _ _ _ _ _ _ (is_preconnected_Ioo)
_ _ (right_nhds_within_Ioo_ne_bot hlt) (left_nhds_within_Ioo_ne_bot hlt)
inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self)
_ _ ((hf.continuous_w... | lemma | intermediate_value_Ioo' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"eq_or_lt_of_le",
"inf_le_right",
"is_preconnected.intermediate_value_Ioo",
"is_preconnected_Ioo",
"left_nhds_within_Ioo_ne_bot",
"right_nhds_within_Ioo_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.surj_on_Icc {s : set α} [hs : ord_connected s] {f : α → δ}
(hf : continuous_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
surj_on f s (Icc (f a) (f b)) | hs.is_preconnected.intermediate_value ha hb hf | lemma | continuous_on.surj_on_Icc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `Icc (f x) (f y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.surj_on_uIcc {s : set α} [hs : ord_connected s] {f : α → δ}
(hf : continuous_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
surj_on f s (uIcc (f a) (f b)) | by cases le_total (f a) (f b) with hab hab; simp [hf.surj_on_Icc, *] | lemma | continuous_on.surj_on_uIcc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.surjective {f : α → δ} (hf : continuous f) (h_top : tendsto f at_top at_top)
(h_bot : tendsto f at_bot at_bot) :
function.surjective f | λ p, mem_range_of_exists_le_of_exists_ge hf
(h_bot.eventually (eventually_le_at_bot p)).exists
(h_top.eventually (eventually_ge_at_top p)).exists | lemma | continuous.surjective | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"mem_range_of_exists_le_of_exists_ge"
] | A continuous function which tendsto `at_top` `at_top` and to `at_bot` `at_bot` is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.surjective' {f : α → δ} (hf : continuous f) (h_top : tendsto f at_bot at_top)
(h_bot : tendsto f at_top at_bot) :
function.surjective f | @continuous.surjective αᵒᵈ _ _ _ _ _ _ _ _ _ hf h_top h_bot | lemma | continuous.surjective' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"continuous.surjective"
] | A continuous function which tendsto `at_bot` `at_top` and to `at_top` `at_bot` is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.surj_on_of_tendsto {f : α → δ} {s : set α} [ord_connected s]
(hs : s.nonempty) (hf : continuous_on f s) (hbot : tendsto (λ x : s, f x) at_bot at_bot)
(htop : tendsto (λ x : s, f x) at_top at_top) :
surj_on f s univ | by haveI := classical.inhabited_of_nonempty hs.to_subtype;
exact (surj_on_iff_surjective.2 $
(continuous_on_iff_continuous_restrict.1 hf).surjective htop hbot) | lemma | continuous_on.surj_on_of_tendsto | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on"
] | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `at_bot : filter β` along `at_bot : filter ↥s` and tends to `at_top : filter β` along
`at_top : filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the
conclusion as `surj_on f s univ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.surj_on_of_tendsto' {f : α → δ} {s : set α} [ord_connected s]
(hs : s.nonempty) (hf : continuous_on f s) (hbot : tendsto (λ x : s, f x) at_bot at_top)
(htop : tendsto (λ x : s, f x) at_top at_bot) :
surj_on f s univ | @continuous_on.surj_on_of_tendsto α _ _ _ _ δᵒᵈ _ _ _ _ _ _ hs hf hbot htop | lemma | continuous_on.surj_on_of_tendsto' | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on.surj_on_of_tendsto"
] | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `at_top : filter β` along `at_bot : filter ↥s` and tends to `at_bot : filter β` along
`at_top : filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the
conclusion as `surj_on f s univ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_Ioi_iff_Ici {a : α} {f : α → β} :
continuous_within_at f (Ioi a) a ↔ continuous_within_at f (Ici a) a | by simp only [← Ici_diff_left, continuous_within_at_diff_self] | lemma | continuous_within_at_Ioi_iff_Ici | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"continuous_within_at",
"continuous_within_at_diff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Iio_iff_Iic {a : α} {f : α → β} :
continuous_within_at f (Iio a) a ↔ continuous_within_at f (Iic a) a | @continuous_within_at_Ioi_iff_Ici αᵒᵈ _ ‹topological_space α› _ _ _ f | lemma | continuous_within_at_Iio_iff_Iic | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"continuous_within_at",
"continuous_within_at_Ioi_iff_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_left'_le_nhds_ne (a : α) :
𝓝[<] a ≤ 𝓝[≠] a | nhds_within_mono a (λ y hy, ne_of_lt hy) | lemma | nhds_left'_le_nhds_ne | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_right'_le_nhds_ne (a : α) :
𝓝[>] a ≤ 𝓝[≠] a | nhds_within_mono a (λ y hy, ne_of_gt hy) | lemma | nhds_right'_le_nhds_ne | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_left_sup_nhds_right (a : α) :
𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a | by rw [← nhds_within_union, Iic_union_Ici, nhds_within_univ] | lemma | nhds_left_sup_nhds_right | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_union",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_left'_sup_nhds_right (a : α) :
𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a | by rw [← nhds_within_union, Iio_union_Ici, nhds_within_univ] | lemma | nhds_left'_sup_nhds_right | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_union",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_left_sup_nhds_right' (a : α) :
𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a | by rw [← nhds_within_union, Iic_union_Ioi, nhds_within_univ] | lemma | nhds_left_sup_nhds_right' | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_union",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_left'_sup_nhds_right' (a : α) :
𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a | by rw [← nhds_within_union, Iio_union_Ioi] | lemma | nhds_left'_sup_nhds_right' | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"nhds_within_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff_continuous_left_right {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iic a) a ∧ continuous_within_at f (Ici a) a | by simp only [continuous_within_at, continuous_at, ← tendsto_sup, nhds_left_sup_nhds_right] | lemma | continuous_at_iff_continuous_left_right | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"continuous_at",
"continuous_within_at",
"nhds_left_sup_nhds_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff_continuous_left'_right' {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iio a) a ∧ continuous_within_at f (Ioi a) a | by rw [continuous_within_at_Ioi_iff_Ici, continuous_within_at_Iio_iff_Iic,
continuous_at_iff_continuous_left_right] | lemma | continuous_at_iff_continuous_left'_right' | topology.algebra.order | src/topology/algebra/order/left_right.lean | [
"topology.continuous_on"
] | [
"continuous_at",
"continuous_at_iff_continuous_left_right",
"continuous_within_at",
"continuous_within_at_Iio_iff_Iic",
"continuous_within_at_Ioi_iff_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.left_lim (f : α → β) (a : α) : β | begin
classical,
haveI : nonempty β := ⟨f a⟩,
letI : topological_space α := preorder.topology α,
exact if (𝓝[<] a = ⊥) ∨ ¬(∃ y, tendsto f (𝓝[<] a) (𝓝 y)) then f a
else lim (𝓝[<] a) f
end | def | function.left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"lim",
"preorder.topology",
"topological_space"
] | Let `f : α → β` be a function from a linear order `α` to a topological_space `β`, and
let `a : α`. The limit strictly to the left of `f` at `a`, denoted with `left_lim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its left or the function has no left
limit, we use `f a` instead to guarantee... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.right_lim (f : α → β) (a : α) : β | @function.left_lim αᵒᵈ β _ _ f a | def | function.right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"function.left_lim"
] | Let `f : α → β` be a function from a linear order `α` to a topological_space `β`, and
let `a : α`. The limit strictly to the right of `f` at `a`, denoted with `right_lim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its right or the function has no right
limit, , we use `f a` instead to gua... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_lim_eq_of_tendsto
[hα : topological_space α] [h'α : order_topology α] [t2_space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : tendsto f (𝓝[<] a) (𝓝 y)) :
left_lim f a = y | begin
have h'' : ∃ y, tendsto f (𝓝[<] a) (𝓝 y) := ⟨y, h'⟩,
rw [h'α.topology_eq_generate_intervals] at h h' h'',
simp only [left_lim, h, h'', not_true, or_self, if_false],
haveI := ne_bot_iff.2 h,
exact h'.lim_eq,
end | lemma | left_lim_eq_of_tendsto | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"order_topology",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_eq_of_eq_bot [hα : topological_space α] [h'α : order_topology α]
(f : α → β) {a : α} (h : 𝓝[<] a = ⊥) :
left_lim f a = f a | begin
rw [h'α.topology_eq_generate_intervals] at h,
simp [left_lim, ite_eq_left_iff, h],
end | lemma | left_lim_eq_of_eq_bot | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"ite_eq_left_iff",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_eq_Sup [topological_space α] [order_topology α] (h : 𝓝[<] x ≠ ⊥) :
left_lim f x = Sup (f '' (Iio x)) | left_lim_eq_of_tendsto h (hf.tendsto_nhds_within_Iio x) | lemma | monotone.left_lim_eq_Sup | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"left_lim_eq_of_tendsto",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_le (h : x ≤ y) : left_lim f x ≤ f y | begin
letI : topological_space α := preorder.topology α,
haveI : order_topology α := ⟨rfl⟩,
rcases eq_or_ne (𝓝[<] x) ⊥ with h'|h',
{ simpa [left_lim, h'] using hf h },
haveI A : ne_bot (𝓝[<] x) := ne_bot_iff.2 h',
rw left_lim_eq_Sup hf h',
refine cSup_le _ _,
{ simp only [nonempty_image_iff],
exac... | lemma | monotone.left_lim_le | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"and_imp",
"cSup_le",
"eq_or_ne",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"order_topology",
"preorder.topology",
"self_mem_nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_left_lim (h : x < y) : f x ≤ left_lim f y | begin
letI : topological_space α := preorder.topology α,
haveI : order_topology α := ⟨rfl⟩,
rcases eq_or_ne (𝓝[<] y) ⊥ with h'|h',
{ rw left_lim_eq_of_eq_bot _ h', exact hf h.le },
rw left_lim_eq_Sup hf h',
refine le_cSup ⟨f y, _⟩ (mem_image_of_mem _ h),
simp only [upper_bounds, mem_image, mem_Iio, foral... | lemma | monotone.le_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"and_imp",
"eq_or_ne",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"le_cSup",
"left_lim_eq_of_eq_bot",
"order_topology",
"preorder.topology",
"topological_space",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim : monotone (left_lim f) | begin
assume x y h,
rcases eq_or_lt_of_le h with rfl|hxy,
{ exact le_rfl },
{ exact (hf.left_lim_le le_rfl).trans (hf.le_left_lim hxy) }
end | lemma | monotone.left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"eq_or_lt_of_le",
"le_rfl",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_right_lim (h : x ≤ y) : f x ≤ right_lim f y | hf.dual.left_lim_le h | lemma | monotone.le_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim_le (h : x < y) : right_lim f x ≤ f y | hf.dual.le_left_lim h | lemma | monotone.right_lim_le | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim : monotone (right_lim f) | λ x y h, hf.dual.left_lim h | lemma | monotone.right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_le_right_lim (h : x ≤ y) : left_lim f x ≤ right_lim f y | (hf.left_lim_le le_rfl).trans (hf.le_right_lim h) | lemma | monotone.left_lim_le_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim_le_left_lim (h : x < y) : right_lim f x ≤ left_lim f y | begin
letI : topological_space α := preorder.topology α,
haveI : order_topology α := ⟨rfl⟩,
rcases eq_or_ne (𝓝[<] y) ⊥ with h'|h',
{ simp [left_lim, h'],
exact right_lim_le hf h },
obtain ⟨a, ⟨xa, ay⟩⟩ : (Ioo x y).nonempty :=
forall_mem_nonempty_iff_ne_bot.2 (ne_bot_iff.2 h') (Ioo x y)
(Ioo_mem... | lemma | monotone.right_lim_le_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"Ioo_mem_nhds_within_Iio",
"eq_or_ne",
"order_topology",
"preorder.topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_left_lim (x : α) : tendsto f (𝓝[<] x) (𝓝 (left_lim f x)) | begin
rcases eq_or_ne (𝓝[<] x) ⊥ with h'|h',
{ simp [h'] },
rw left_lim_eq_Sup hf h',
exact hf.tendsto_nhds_within_Iio x
end | lemma | monotone.tendsto_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_left_lim_within (x : α) : tendsto f (𝓝[<] x) (𝓝[≤] (left_lim f x)) | begin
apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within f (hf.tendsto_left_lim x),
filter_upwards [self_mem_nhds_within] with y hy using hf.le_left_lim hy,
end | lemma | monotone.tendsto_left_lim_within | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"self_mem_nhds_within",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_right_lim (x : α) :
tendsto f (𝓝[>] x) (𝓝 (right_lim f x)) | hf.dual.tendsto_left_lim x | lemma | monotone.tendsto_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_right_lim_within (x : α) :
tendsto f (𝓝[>] x) (𝓝[≥] (right_lim f x)) | hf.dual.tendsto_left_lim_within x | lemma | monotone.tendsto_right_lim_within | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Iio_iff_left_lim_eq :
continuous_within_at f (Iio x) x ↔ left_lim f x = f x | begin
rcases eq_or_ne (𝓝[<] x) ⊥ with h'|h',
{ simp [left_lim_eq_of_eq_bot f h', continuous_within_at, h'] },
haveI : (𝓝[Iio x] x).ne_bot := ne_bot_iff.2 h',
refine ⟨λ h, tendsto_nhds_unique (hf.tendsto_left_lim x) h.tendsto, λ h, _⟩,
have := hf.tendsto_left_lim x,
rwa h at this,
end | lemma | monotone.continuous_within_at_Iio_iff_left_lim_eq | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_within_at",
"eq_or_ne",
"left_lim_eq_of_eq_bot",
"tendsto_nhds_unique"
] | A monotone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_Ioi_iff_right_lim_eq :
continuous_within_at f (Ioi x) x ↔ right_lim f x = f x | hf.dual.continuous_within_at_Iio_iff_left_lim_eq | lemma | monotone.continuous_within_at_Ioi_iff_right_lim_eq | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_within_at"
] | A monotone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_iff_left_lim_eq_right_lim :
continuous_at f x ↔ left_lim f x = right_lim f x | begin
refine ⟨λ h, _, λ h, _⟩,
{ have A : left_lim f x = f x,
from (hf.continuous_within_at_Iio_iff_left_lim_eq).1 h.continuous_within_at,
have B : right_lim f x = f x,
from (hf.continuous_within_at_Ioi_iff_right_lim_eq).1 h.continuous_within_at,
exact A.trans B.symm },
{ have h' : left_lim f ... | lemma | monotone.continuous_at_iff_left_lim_eq_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_at"
] | A monotone function is continuous at a point if and only if its left and right limits
coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_not_continuous_within_at_Ioi [topological_space.second_countable_topology β] :
set.countable {x | ¬(continuous_within_at f (Ioi x) x)} | begin
/- If `f` is not continuous on the right at `x`, there is an interval `(f x, z x)` which is not
reached by `f`. This gives a family of disjoint open intervals in `β`. Such a family can only
be countable as `β` is second-countable. -/
nontriviality α,
let s := {x | ¬(continuous_within_at f (Ioi x) x)},
... | lemma | monotone.countable_not_continuous_within_at_Ioi | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"Ioo_mem_nhds_within_Ioi",
"continuous_within_at",
"countable",
"ne_of_apply_ne",
"self_mem_nhds_within",
"set.countable",
"set.pairwise_disjoint.countable_of_Ioo",
"strict_mono_on.inj_on",
"topological_space.second_countable_topology"
] | In a second countable space, the set of points where a monotone function is not right-continuous
is at most countable. Superseded by `countable_not_continuous_at` which gives the two-sided
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_not_continuous_within_at_Iio [topological_space.second_countable_topology β] :
set.countable {x | ¬(continuous_within_at f (Iio x) x)} | hf.dual.countable_not_continuous_within_at_Ioi | lemma | monotone.countable_not_continuous_within_at_Iio | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_within_at",
"set.countable",
"topological_space.second_countable_topology"
] | In a second countable space, the set of points where a monotone function is not left-continuous
is at most countable. Superseded by `countable_not_continuous_at` which gives the two-sided
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_not_continuous_at [topological_space.second_countable_topology β] :
set.countable {x | ¬(continuous_at f x)} | begin
apply (hf.countable_not_continuous_within_at_Ioi.union
hf.countable_not_continuous_within_at_Iio).mono _,
refine compl_subset_compl.1 _,
simp only [compl_union],
rintros x ⟨hx, h'x⟩,
simp only [mem_set_of_eq, not_not, mem_compl_iff] at hx h'x ⊢,
exact continuous_at_iff_continuous_left'_right'... | lemma | monotone.countable_not_continuous_at | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"not_not",
"set.countable",
"topological_space.second_countable_topology"
] | In a second countable space, the set of points where a monotone function is not continuous
is at most countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_left_lim (h : x ≤ y) : f y ≤ left_lim f x | hf.dual_right.left_lim_le h | lemma | antitone.le_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_le (h : x < y) : left_lim f y ≤ f x | hf.dual_right.le_left_lim h | lemma | antitone.left_lim_le | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim : antitone (left_lim f) | hf.dual_right.left_lim | lemma | antitone.left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim_le (h : x ≤ y) : right_lim f y ≤ f x | hf.dual_right.le_right_lim h | lemma | antitone.right_lim_le | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_right_lim (h : x < y) : f y ≤ right_lim f x | hf.dual_right.right_lim_le h | lemma | antitone.le_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim : antitone (right_lim f) | hf.dual_right.right_lim | lemma | antitone.right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_lim_le_left_lim (h : x ≤ y) : right_lim f y ≤ left_lim f x | hf.dual_right.left_lim_le_right_lim h | lemma | antitone.right_lim_le_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lim_le_right_lim (h : x < y) : left_lim f y ≤ right_lim f x | hf.dual_right.right_lim_le_left_lim h | lemma | antitone.left_lim_le_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_left_lim (x : α) : tendsto f (𝓝[<] x) (𝓝 (left_lim f x)) | hf.dual_right.tendsto_left_lim x | lemma | antitone.tendsto_left_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_left_lim_within (x : α) : tendsto f (𝓝[<] x) (𝓝[≥] (left_lim f x)) | hf.dual_right.tendsto_left_lim_within x | lemma | antitone.tendsto_left_lim_within | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_right_lim (x : α) :
tendsto f (𝓝[>] x) (𝓝 (right_lim f x)) | hf.dual_right.tendsto_right_lim x | lemma | antitone.tendsto_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_right_lim_within (x : α) :
tendsto f (𝓝[>] x) (𝓝[≤] (right_lim f x)) | hf.dual_right.tendsto_right_lim_within x | lemma | antitone.tendsto_right_lim_within | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_Iio_iff_left_lim_eq :
continuous_within_at f (Iio x) x ↔ left_lim f x = f x | hf.dual_right.continuous_within_at_Iio_iff_left_lim_eq | lemma | antitone.continuous_within_at_Iio_iff_left_lim_eq | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_within_at"
] | An antitone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_Ioi_iff_right_lim_eq :
continuous_within_at f (Ioi x) x ↔ right_lim f x = f x | hf.dual_right.continuous_within_at_Ioi_iff_right_lim_eq | lemma | antitone.continuous_within_at_Ioi_iff_right_lim_eq | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_within_at"
] | An antitone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_iff_left_lim_eq_right_lim :
continuous_at f x ↔ left_lim f x = right_lim f x | hf.dual_right.continuous_at_iff_left_lim_eq_right_lim | lemma | antitone.continuous_at_iff_left_lim_eq_right_lim | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_at"
] | An antitone function is continuous at a point if and only if its left and right limits
coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_not_continuous_at [topological_space.second_countable_topology β] :
set.countable {x | ¬(continuous_at f x)} | hf.dual_right.countable_not_continuous_at | lemma | antitone.countable_not_continuous_at | topology.algebra.order | src/topology/algebra/order/left_right_lim.lean | [
"topology.order.basic",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"set.countable",
"topological_space.second_countable_topology"
] | In a second countable space, the set of points where an antitone function is not continuous
is at most countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_le_nhds_class (α : Type*) [preorder α] [topological_space α] : Prop | (is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤)) | class | 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"
] | [
"is_bounded_le_nhds",
"topological_space"
] | Ad hoc typeclass stating that neighborhoods are eventually bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_ge_nhds_class (α : Type*) [preorder α] [topological_space α] : Prop | (is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥)) | class | 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"
] | [
"is_bounded_ge_nhds",
"topological_space"
] | Ad hoc typeclass stating that neighborhoods are eventually bounded below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤) | bounded_le_nhds_class.is_bounded_le_nhds _ | lemma | is_bounded_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.is_bounded_under_le (h : tendsto u f (𝓝 a)) :
f.is_bounded_under (≤) u | (is_bounded_le_nhds a).mono h | lemma | filter.tendsto.is_bounded_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"
] | [
"is_bounded_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.bdd_above_range_of_cofinite [is_directed α (≤)]
(h : tendsto u cofinite (𝓝 a)) : bdd_above (set.range u) | h.is_bounded_under_le.bdd_above_range_of_cofinite | lemma | filter.tendsto.bdd_above_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_above",
"is_directed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.bdd_above_range [is_directed α (≤)] {u : ℕ → α} (h : tendsto u at_top (𝓝 a)) :
bdd_above (set.range u) | h.is_bounded_under_le.bdd_above_range | lemma | filter.tendsto.bdd_above_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_above",
"is_directed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_ge_nhds (a : α) : (𝓝 a).is_cobounded (≥) | (is_bounded_le_nhds a).is_cobounded_flip | lemma | is_cobounded_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"
] | [
"is_bounded_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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