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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Union_union_compl_to_subordinate_boxes (π₁ : tagged_prepartition I)
(π₂ : prepartition I) (hU : π₂.Union = I \ π₁.Union) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π₁.union_compl_to_subordinate π₂ hU r).Union = I | (is_partition_union_compl_to_subordinate _ _ _ _).Union_eq | lemma | box_integral.tagged_prepartition.Union_union_compl_to_subordinate_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_union_compl_to_subordinate (π₁ : tagged_prepartition I)
(π₂ : prepartition I) (hU : π₂.Union = I \ π₁.Union) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π₁.union_compl_to_subordinate π₂ hU r).distortion = max π₁.distortion π₂.distortion | by simp [union_compl_to_subordinate] | lemma | box_integral.tagged_prepartition.distortion_union_compl_to_subordinate | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tagged_prepartition (I : box ι) extends prepartition I | (tag : box ι → ι → ℝ)
(tag_mem_Icc : ∀ J, tag J ∈ I.Icc) | structure | box_integral.tagged_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | A tagged prepartition is a prepartition enriched with a tagged point for each box of the
prepartition. For simiplicity we require that `tag` is defined for all boxes in `ι → ℝ` but
we will use onle the values of `tag` on the boxes of the partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_prepartition {π : tagged_prepartition I} :
J ∈ π.to_prepartition ↔ J ∈ π | iff.rfl | lemma | box_integral.tagged_prepartition.mem_to_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk (π : prepartition I) (f h) :
J ∈ mk π f h ↔ J ∈ π | iff.rfl | lemma | box_integral.tagged_prepartition.mem_mk | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union : set (ι → ℝ) | π.to_prepartition.Union | def | box_integral.tagged_prepartition.Union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | Union of all boxes of a tagged prepartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Union_mk (π : prepartition I) (f h) : (mk π f h).Union = π.Union | rfl | lemma | box_integral.tagged_prepartition.Union_mk | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_to_prepartition : π.to_prepartition.Union = π.Union | rfl | lemma | box_integral.tagged_prepartition.Union_to_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition | π.to_prepartition.is_partition | def | box_integral.tagged_prepartition.is_partition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | A tagged prepartition is a partition if it covers the whole box. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_partition_iff_Union_eq : is_partition π ↔ π.Union = I | prepartition.is_partition_iff_Union_eq | lemma | box_integral.tagged_prepartition.is_partition_iff_Union_eq | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter (p : box ι → Prop) : tagged_prepartition I | ⟨π.1.filter p, π.2, π.3⟩ | def | box_integral.tagged_prepartition.filter | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"filter"
] | The tagged partition made of boxes of `π` that satisfy predicate `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_filter {p : box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J | finset.mem_filter | lemma | box_integral.tagged_prepartition.mem_filter | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset.mem_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_filter_not (π : tagged_prepartition I) (p : box ι → Prop) :
(π.filter (λ J, ¬p J)).Union = π.Union \ (π.filter p).Union | π.to_prepartition.Union_filter_not p | lemma | box_integral.tagged_prepartition.Union_filter_not | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_tagged (π : prepartition I) (πi : Π J, tagged_prepartition J) :
tagged_prepartition I | { to_prepartition := π.bUnion (λ J, (πi J).to_prepartition),
tag := λ J, (πi (π.bUnion_index (λ J, (πi J).to_prepartition) J)).tag J,
tag_mem_Icc := λ J, box.le_iff_Icc.1 (π.bUnion_index_le _ _) ((πi _).tag_mem_Icc _) } | def | box_integral.prepartition.bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | Given a partition `π` of `I : box_integral.box ι` and a collection of tagged partitions
`πi J` of all boxes `J ∈ π`, returns the tagged partition of `I` into all the boxes of `πi J`
with tags coming from `(πi J).tag`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_bUnion_tagged (π : prepartition I) {πi : Π J, tagged_prepartition J} :
J ∈ π.bUnion_tagged πi ↔ ∃ J' ∈ π, J ∈ πi J' | π.mem_bUnion | lemma | box_integral.prepartition.mem_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tag_bUnion_tagged (π : prepartition I) {πi : Π J, tagged_prepartition J} (hJ : J ∈ π) {J'}
(hJ' : J' ∈ πi J) :
(π.bUnion_tagged πi).tag J' = (πi J).tag J' | begin
have : J' ∈ π.bUnion_tagged πi, from π.mem_bUnion.2 ⟨J, hJ, hJ'⟩,
obtain rfl := π.bUnion_index_of_mem hJ hJ',
refl
end | lemma | box_integral.prepartition.tag_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_bUnion_tagged (π : prepartition I) (πi : Π J, tagged_prepartition J) :
(π.bUnion_tagged πi).Union = ⋃ J ∈ π, (πi J).Union | Union_bUnion _ _ | lemma | box_integral.prepartition.Union_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_bUnion_tagged (p : (ι → ℝ) → box ι → Prop) (π : prepartition I)
(πi : Π J, tagged_prepartition J) :
(∀ J ∈ π.bUnion_tagged πi, p ((π.bUnion_tagged πi).tag J) J) ↔
∀ (J ∈ π) (J' ∈ πi J), p ((πi J).tag J') J' | begin
simp only [bex_imp_distrib, mem_bUnion_tagged],
refine ⟨λ H J hJ J' hJ', _, λ H J' J hJ hJ', _⟩,
{ rw ← π.tag_bUnion_tagged hJ hJ', exact H J' J hJ hJ' },
{ rw π.tag_bUnion_tagged hJ hJ', exact H J hJ J' hJ' }
end | lemma | box_integral.prepartition.forall_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"bex_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition.bUnion_tagged {π : prepartition I} (h : is_partition π)
{πi : Π J, tagged_prepartition J} (hi : ∀ J ∈ π, (πi J).is_partition) :
(π.bUnion_tagged πi).is_partition | h.bUnion hi | lemma | box_integral.prepartition.is_partition.bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_prepartition (π : tagged_prepartition I) (πi : Π J, prepartition J) :
tagged_prepartition I | { to_prepartition := π.to_prepartition.bUnion πi,
tag := λ J, π.tag (π.to_prepartition.bUnion_index πi J),
tag_mem_Icc := λ J, π.tag_mem_Icc _ } | def | box_integral.tagged_prepartition.bUnion_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | Given a tagged partition `π` of `I` and a (not tagged) partition `πi J hJ` of each `J ∈ π`,
returns the tagged partition of `I` into all the boxes of all `πi J hJ`. The tag of a box `J`
is defined to be the `π.tag` of the box of the partition `π` that includes `J`.
Note that usually the result is not a Henstock partit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_partition.bUnion_prepartition {π : tagged_prepartition I} (h : is_partition π)
{πi : Π J, prepartition J} (hi : ∀ J ∈ π, (πi J).is_partition) :
(π.bUnion_prepartition πi).is_partition | h.bUnion hi | lemma | box_integral.tagged_prepartition.is_partition.bUnion_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_prepartition (π : tagged_prepartition I) (π' : prepartition I) :
tagged_prepartition I | π.bUnion_prepartition $ λ J, π'.restrict J | def | box_integral.tagged_prepartition.inf_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | Given two partitions `π₁` and `π₁`, one of them tagged and the other is not, returns the tagged
partition with `to_partition = π₁.to_partition ⊓ π₂` and tags coming from `π₁`.
Note that usually the result is not a Henstock partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_prepartition_to_prepartition (π : tagged_prepartition I) (π' : prepartition I) :
(π.inf_prepartition π').to_prepartition = π.to_prepartition ⊓ π' | rfl | lemma | box_integral.tagged_prepartition.inf_prepartition_to_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf_prepartition_comm :
J ∈ π₁.inf_prepartition π₂.to_prepartition ↔ J ∈ π₂.inf_prepartition π₁.to_prepartition | by simp only [← mem_to_prepartition, inf_prepartition_to_prepartition, inf_comm] | lemma | box_integral.tagged_prepartition.mem_inf_prepartition_comm | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition.inf_prepartition (h₁ : π₁.is_partition) {π₂ : prepartition I}
(h₂ : π₂.is_partition) :
(π₁.inf_prepartition π₂).is_partition | h₁.inf h₂ | lemma | box_integral.tagged_prepartition.is_partition.inf_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Henstock (π : tagged_prepartition I) : Prop | ∀ J ∈ π, π.tag J ∈ J.Icc | def | box_integral.tagged_prepartition.is_Henstock | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | A tagged partition is said to be a Henstock partition if for each `J ∈ π`, the tag of `J`
belongs to `J.Icc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Henstock_bUnion_tagged
{π : prepartition I} {πi : Π J, tagged_prepartition J} :
is_Henstock (π.bUnion_tagged πi) ↔ ∀ J ∈ π, (πi J).is_Henstock | π.forall_bUnion_tagged (λ x J, x ∈ J.Icc) πi | lemma | box_integral.tagged_prepartition.is_Henstock_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Henstock.card_filter_tag_eq_le [fintype ι] (h : π.is_Henstock) (x : ι → ℝ) :
(π.boxes.filter (λ J, π.tag J = x)).card ≤ 2 ^ fintype.card ι | calc (π.boxes.filter (λ J, π.tag J = x)).card ≤ (π.boxes.filter (λ J : box ι, x ∈ J.Icc)).card :
begin
refine finset.card_le_of_subset (λ J hJ, _),
rw finset.mem_filter at hJ ⊢, rcases hJ with ⟨hJ, rfl⟩,
exact ⟨hJ, h J hJ⟩
end
... ≤ 2 ^ fintype.card ι : π.to_prepartition.card_filter_mem_Icc_le x | lemma | box_integral.tagged_prepartition.is_Henstock.card_filter_tag_eq_le | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset.card_le_of_subset",
"finset.mem_filter",
"fintype",
"fintype.card"
] | In a Henstock prepartition, there are at most `2 ^ fintype.card ι` boxes with a given tag. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subordinate [fintype ι] (π : tagged_prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) : Prop | ∀ J ∈ π, (J : _).Icc ⊆ closed_ball (π.tag J) (r $ π.tag J) | def | box_integral.tagged_prepartition.is_subordinate | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | A tagged partition `π` is subordinate to `r : (ι → ℝ) → ℝ` if each box `J ∈ π` is included in
the closed ball with center `π.tag J` and radius `r (π.tag J)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_subordinate_bUnion_tagged [fintype ι]
{π : prepartition I} {πi : Π J, tagged_prepartition J} :
is_subordinate (π.bUnion_tagged πi) r ↔ ∀ J ∈ π, (πi J).is_subordinate r | π.forall_bUnion_tagged (λ x J, J.Icc ⊆ closed_ball x (r x)) πi | lemma | box_integral.tagged_prepartition.is_subordinate_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.bUnion_prepartition [fintype ι] (h : is_subordinate π r)
(πi : Π J, prepartition J) :
is_subordinate (π.bUnion_prepartition πi) r | λ J hJ, subset.trans (box.le_iff_Icc.1 $ π.to_prepartition.le_bUnion_index hJ) $
h _ $ π.to_prepartition.bUnion_index_mem hJ | lemma | box_integral.tagged_prepartition.is_subordinate.bUnion_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.inf_prepartition [fintype ι] (h : is_subordinate π r) (π' : prepartition I) :
is_subordinate (π.inf_prepartition π') r | h.bUnion_prepartition _ | lemma | box_integral.tagged_prepartition.is_subordinate.inf_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.mono' [fintype ι] {π : tagged_prepartition I}
(hr₁ : π.is_subordinate r₁) (h : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J)) :
π.is_subordinate r₂ | λ J hJ x hx, closed_ball_subset_closed_ball (h _ hJ) (hr₁ _ hJ hx) | lemma | box_integral.tagged_prepartition.is_subordinate.mono' | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.mono [fintype ι] {π : tagged_prepartition I}
(hr₁ : π.is_subordinate r₁) (h : ∀ x ∈ I.Icc, r₁ x ≤ r₂ x) :
π.is_subordinate r₂ | hr₁.mono' $ λ J _, h _ $ π.tag_mem_Icc J | lemma | box_integral.tagged_prepartition.is_subordinate.mono | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.diam_le [fintype ι] {π : tagged_prepartition I}
(h : π.is_subordinate r) (hJ : J ∈ π.boxes) :
diam J.Icc ≤ 2 * r (π.tag J) | calc diam J.Icc ≤ diam (closed_ball (π.tag J) (r $ π.tag J)) :
diam_mono (h J hJ) bounded_closed_ball
... ≤ 2 * r (π.tag J) : diam_closed_ball (le_of_lt (r _).2) | lemma | box_integral.tagged_prepartition.is_subordinate.diam_le | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (I J : box ι) (hJ : J ≤ I) (x : ι → ℝ) (h : x ∈ I.Icc) : tagged_prepartition I | ⟨prepartition.single I J hJ, λ J, x, λ J, h⟩ | def | box_integral.tagged_prepartition.single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | Tagged prepartition with single box and prescribed tag. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_single {J'} (hJ : J ≤ I) (h : x ∈ I.Icc) : J' ∈ single I J hJ x h ↔ J' = J | finset.mem_singleton | lemma | box_integral.tagged_prepartition.mem_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_single_iff (hJ : J ≤ I) (h : x ∈ I.Icc) :
(single I J hJ x h).is_partition ↔ J = I | prepartition.is_partition_single_iff hJ | lemma | box_integral.tagged_prepartition.is_partition_single_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_single (h : x ∈ I.Icc) : (single I I le_rfl x h).is_partition | prepartition.is_partition_top I | lemma | box_integral.tagged_prepartition.is_partition_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_mem_single (p : (ι → ℝ) → (box ι) → Prop) (hJ : J ≤ I) (h : x ∈ I.Icc) :
(∀ J' ∈ single I J hJ x h, p ((single I J hJ x h).tag J') J') ↔ p x J | by simp | lemma | box_integral.tagged_prepartition.forall_mem_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Henstock_single_iff (hJ : J ≤ I) (h : x ∈ I.Icc) :
is_Henstock (single I J hJ x h) ↔ x ∈ J.Icc | forall_mem_single (λ x J, x ∈ J.Icc) hJ h | lemma | box_integral.tagged_prepartition.is_Henstock_single_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Henstock_single (h : x ∈ I.Icc) : is_Henstock (single I I le_rfl x h) | (is_Henstock_single_iff (le_refl I) h).2 h | lemma | box_integral.tagged_prepartition.is_Henstock_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate_single [fintype ι] (hJ : J ≤ I) (h : x ∈ I.Icc) :
is_subordinate (single I J hJ x h) r ↔ J.Icc ⊆ closed_ball x (r x) | forall_mem_single (λ x J, J.Icc ⊆ closed_ball x (r x)) hJ h | lemma | box_integral.tagged_prepartition.is_subordinate_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_single (hJ : J ≤ I) (h : x ∈ I.Icc) :
(single I J hJ x h).Union = J | prepartition.Union_single hJ | lemma | box_integral.tagged_prepartition.Union_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disj_union (π₁ π₂ : tagged_prepartition I) (h : disjoint π₁.Union π₂.Union) :
tagged_prepartition I | { to_prepartition := π₁.to_prepartition.disj_union π₂.to_prepartition h,
tag := π₁.boxes.piecewise π₁.tag π₂.tag,
tag_mem_Icc := λ J, by { dunfold finset.piecewise, split_ifs,
exacts [π₁.tag_mem_Icc J, π₂.tag_mem_Icc J] } } | def | box_integral.tagged_prepartition.disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"finset.piecewise"
] | Union of two tagged prepartitions with disjoint unions of boxes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disj_union_boxes (h : disjoint π₁.Union π₂.Union) :
(π₁.disj_union π₂ h).boxes = π₁.boxes ∪ π₂.boxes | rfl | lemma | box_integral.tagged_prepartition.disj_union_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_disj_union (h : disjoint π₁.Union π₂.Union) :
J ∈ π₁.disj_union π₂ h ↔ J ∈ π₁ ∨ J ∈ π₂ | finset.mem_union | lemma | box_integral.tagged_prepartition.mem_disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"finset.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_disj_union (h : disjoint π₁.Union π₂.Union) :
(π₁.disj_union π₂ h).Union = π₁.Union ∪ π₂.Union | prepartition.Union_disj_union _ | lemma | box_integral.tagged_prepartition.Union_disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disj_union_tag_of_mem_left (h : disjoint π₁.Union π₂.Union) (hJ : J ∈ π₁) :
(π₁.disj_union π₂ h).tag J = π₁.tag J | dif_pos hJ | lemma | box_integral.tagged_prepartition.disj_union_tag_of_mem_left | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disj_union_tag_of_mem_right (h : disjoint π₁.Union π₂.Union) (hJ : J ∈ π₂) :
(π₁.disj_union π₂ h).tag J = π₂.tag J | dif_neg $ λ h₁, h.le_bot ⟨π₁.subset_Union h₁ J.upper_mem, π₂.subset_Union hJ J.upper_mem⟩ | lemma | box_integral.tagged_prepartition.disj_union_tag_of_mem_right | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.disj_union [fintype ι] (h₁ : is_subordinate π₁ r)
(h₂ : is_subordinate π₂ r) (h : disjoint π₁.Union π₂.Union) :
is_subordinate (π₁.disj_union π₂ h) r | begin
refine λ J hJ, (finset.mem_union.1 hJ).elim (λ hJ, _) (λ hJ, _),
{ rw disj_union_tag_of_mem_left _ hJ, exact h₁ _ hJ },
{ rw disj_union_tag_of_mem_right _ hJ, exact h₂ _ hJ }
end | lemma | box_integral.tagged_prepartition.is_subordinate.disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Henstock.disj_union (h₁ : is_Henstock π₁) (h₂ : is_Henstock π₂)
(h : disjoint π₁.Union π₂.Union) :
is_Henstock (π₁.disj_union π₂ h) | begin
refine λ J hJ, (finset.mem_union.1 hJ).elim (λ hJ, _) (λ hJ, _),
{ rw disj_union_tag_of_mem_left _ hJ, exact h₁ _ hJ },
{ rw disj_union_tag_of_mem_right _ hJ, exact h₂ _ hJ }
end | lemma | box_integral.tagged_prepartition.is_Henstock.disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embed_box (I J : box ι) (h : I ≤ J) :
tagged_prepartition I ↪ tagged_prepartition J | { to_fun := λ π,
{ le_of_mem' := λ J' hJ', (π.le_of_mem' J' hJ').trans h,
tag_mem_Icc := λ J, box.le_iff_Icc.1 h (π.tag_mem_Icc J),
.. π },
inj' := by { rintro ⟨⟨b₁, h₁le, h₁d⟩, t₁, ht₁⟩ ⟨⟨b₂, h₂le, h₂d⟩, t₂, ht₂⟩ H, simpa using H } } | def | box_integral.tagged_prepartition.embed_box | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | If `I ≤ J`, then every tagged prepartition of `I` is a tagged prepartition of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distortion : ℝ≥0 | π.to_prepartition.distortion | def | box_integral.tagged_prepartition.distortion | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | The distortion of a tagged prepartition is the maximum of distortions of its boxes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.box_integral.prepartition.distortion_bUnion_tagged (π : prepartition I)
(πi : Π J, tagged_prepartition J) :
(π.bUnion_tagged πi).distortion = π.boxes.sup (λ J, (πi J).distortion) | sup_bUnion _ _ | lemma | box_integral.prepartition.distortion_bUnion_tagged | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_bUnion_prepartition (π : tagged_prepartition I)
(πi : Π J, prepartition J) :
(π.bUnion_prepartition πi).distortion = π.boxes.sup (λ J, (πi J).distortion) | sup_bUnion _ _ | lemma | box_integral.tagged_prepartition.distortion_bUnion_prepartition | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_single (hJ : J ≤ I) (h : x ∈ I.Icc) :
distortion (single I J hJ x h) = J.distortion | sup_singleton | lemma | box_integral.tagged_prepartition.distortion_single | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_filter_le (p : box ι → Prop) : (π.filter p).distortion ≤ π.distortion | sup_mono (filter_subset _ _) | lemma | box_integral.tagged_prepartition.distortion_filter_le | analysis.box_integral.partition | src/analysis/box_integral/partition/tagged.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff {n : ℕ∞} (f : V →A[𝕜] W) :
cont_diff 𝕜 n f | begin
rw f.decomp,
apply f.cont_linear.cont_diff.add,
simp only,
exact cont_diff_const,
end | lemma | continuous_affine_map.cont_diff | analysis.calculus | src/analysis/calculus/affine_map.lean | [
"analysis.normed_space.continuous_affine_map",
"analysis.calculus.cont_diff"
] | [
"cont_diff",
"cont_diff_const"
] | A continuous affine map between normed vector spaces is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_smooth_tsupport_subset {s : set E} {x : E} (hs : s ∈ 𝓝 x) :
∃ (f : E → ℝ), tsupport f ⊆ s ∧ has_compact_support f ∧ cont_diff ℝ ⊤ f ∧
range f ⊆ Icc 0 1 ∧ f x = 1 | begin
obtain ⟨d, d_pos, hd⟩ : ∃ (d : ℝ) (hr : 0 < d), euclidean.closed_ball x d ⊆ s,
from euclidean.nhds_basis_closed_ball.mem_iff.1 hs,
let c : cont_diff_bump (to_euclidean x) :=
{ r := d/2,
R := d,
r_pos := half_pos d_pos,
r_lt_R := half_lt_self d_pos },
let f : E → ℝ := c ∘ to_euclidean,
ha... | theorem | exists_smooth_tsupport_subset | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"closure_mono",
"cont_diff",
"cont_diff_bump",
"continuous_linear_equiv.cont_diff",
"euclidean.ball",
"euclidean.ball_subset_closed_ball",
"euclidean.closed_ball",
"euclidean.closure_ball",
"euclidean.is_closed_closed_ball",
"function.support",
"half_pos",
"is_closed.closure_subset_iff",
"is... | If a set `s` is a neighborhood of `x`, then there exists a smooth function `f` taking
values in `[0, 1]`, supported in `s` and with `f x = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open.exists_smooth_support_eq {s : set E} (hs : is_open s) :
∃ (f : E → ℝ), f.support = s ∧ cont_diff ℝ ⊤ f ∧ set.range f ⊆ set.Icc 0 1 | begin
/- For any given point `x` in `s`, one can construct a smooth function with support in `s` and
nonzero at `x`. By second-countability, it follows that we may cover `s` with the supports of
countably many such functions, say `g i`.
Then `∑ i, r i • g i` will be the desired function if `r i` is a sequence o... | theorem | is_open.exists_smooth_support_eq | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"algebra.id.smul_eq_mul",
"bdd_above",
"cont_diff",
"cont_diff_const",
"cont_diff_tsum_of_eventually",
"exists_smooth_tsupport_subset",
"filter.eventually_at_top",
"finset.le_max'",
"finset.mem_image_of_mem",
"finset.mem_range",
"finset.range",
"ge_iff_le",
"has_sum",
"is_open",
"iterate... | Given an open set `s` in a finite-dimensional real normed vector space, there exists a smooth
function with values in `[0, 1]` whose support is exactly `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
φ : E → ℝ | (closed_ball (0 : E) 1).indicator (λ y, (1 : ℝ)) | def | exists_cont_diff_bump_base.φ | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | An auxiliary function to construct partitions of unity on finite-dimensional real vector spaces.
It is the characteristic function of the closed unit ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
u_exists : ∃ u : E → ℝ, cont_diff ℝ ⊤ u ∧
(∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ (support u = ball 0 1) ∧ (∀ x, u (-x) = u x) | begin
have A : is_open (ball (0 : E) 1), from is_open_ball,
obtain ⟨f, f_support, f_smooth, f_range⟩ :
∃ (f : E → ℝ), f.support = ball (0 : E) 1 ∧ cont_diff ℝ ⊤ f ∧ set.range f ⊆ set.Icc 0 1,
from A.exists_smooth_support_eq,
have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := λ x, f_range (mem_range_self x),
refine ⟨λ... | lemma | exists_cont_diff_bump_base.u_exists | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"cont_diff",
"cont_diff_neg",
"is_open",
"not_not",
"set.Icc",
"set.range",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u (x : E) : ℝ | classical.some (u_exists E) x | def | exists_cont_diff_bump_base.u | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | An auxiliary function to construct partitions of unity on finite-dimensional real vector spaces,
which is smooth, symmetric, and with support equal to the unit ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
u_smooth : cont_diff ℝ ⊤ (u : E → ℝ) | (classical.some_spec (u_exists E)).1 | lemma | exists_cont_diff_bump_base.u_smooth | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_continuous : continuous (u : E → ℝ) | (u_smooth E).continuous | lemma | exists_cont_diff_bump_base.u_continuous | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_support : support (u : E → ℝ) = ball 0 1 | (classical.some_spec (u_exists E)).2.2.1 | lemma | exists_cont_diff_bump_base.u_support | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_compact_support : has_compact_support (u : E → ℝ) | begin
rw [has_compact_support_def, u_support, closure_ball (0 : E) one_ne_zero],
exact is_compact_closed_ball _ _,
end | lemma | exists_cont_diff_bump_base.u_compact_support | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"closure_ball",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_nonneg (x : E) : 0 ≤ u x | ((classical.some_spec (u_exists E)).2.1 x).1 | lemma | exists_cont_diff_bump_base.u_nonneg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_le_one (x : E) : u x ≤ 1 | ((classical.some_spec (u_exists E)).2.1 x).2 | lemma | exists_cont_diff_bump_base.u_le_one | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_neg (x : E) : u (-x) = u x | (classical.some_spec (u_exists E)).2.2.2 x | lemma | exists_cont_diff_bump_base.u_neg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
u_int_pos : 0 < ∫ (x : E), u x ∂μ | begin
refine (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _,
{ exact (u_continuous E).integrable_of_has_compact_support (u_compact_support E) },
{ rw u_support, exact measure_ball_pos _ _ zero_lt_one }
end | lemma | exists_cont_diff_bump_base.u_int_pos | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W (D : ℝ) (x : E) : ℝ | ((∫ (x : E), u x ∂μ) * |D|^(finrank ℝ E))⁻¹ • u (D⁻¹ • x) | def | exists_cont_diff_bump_base.W | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | An auxiliary function to construct partitions of unity on finite-dimensional real vector spaces,
which is smooth, symmetric, with support equal to the ball of radius `D` and integral `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
W_def (D : ℝ) :
(W D : E → ℝ) = λ x, ((∫ (x : E), u x ∂μ) * |D|^(finrank ℝ E))⁻¹ • u (D⁻¹ • x) | by { ext1 x, refl } | lemma | exists_cont_diff_bump_base.W_def | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W_nonneg (D : ℝ) (x : E) : 0 ≤ W D x | begin
apply mul_nonneg _ (u_nonneg _),
apply inv_nonneg.2,
apply mul_nonneg (u_int_pos E).le,
apply pow_nonneg (abs_nonneg D)
end | lemma | exists_cont_diff_bump_base.W_nonneg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"abs_nonneg",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W_mul_φ_nonneg (D : ℝ) (x y : E) : 0 ≤ W D y * φ (x - y) | mul_nonneg (W_nonneg D y) (indicator_nonneg (by simp only [zero_le_one, implies_true_iff]) _) | lemma | exists_cont_diff_bump_base.W_mul_φ_nonneg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W_integral {D : ℝ} (Dpos : 0 < D) : ∫ (x : E), W D x ∂μ = 1 | begin
simp_rw [W, integral_smul],
rw [integral_comp_inv_smul_of_nonneg μ (u : E → ℝ) Dpos.le,
abs_of_nonneg Dpos.le, mul_comm],
field_simp [Dpos.ne', (u_int_pos E).ne'],
end | lemma | exists_cont_diff_bump_base.W_integral | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"abs_of_nonneg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W_support {D : ℝ} (Dpos : 0 < D) : support (W D : E → ℝ) = ball 0 D | begin
have B : D • ball (0 : E) 1 = ball 0 D,
by rw [smul_unit_ball Dpos.ne', real.norm_of_nonneg Dpos.le],
have C : D ^ finrank ℝ E ≠ 0, from pow_ne_zero _ Dpos.ne',
simp only [W_def, algebra.id.smul_eq_mul, support_mul, support_inv, univ_inter,
support_comp_inv_smul₀ Dpos.ne', u_support, B, support_cons... | lemma | exists_cont_diff_bump_base.W_support | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"abs_of_nonneg",
"algebra.id.smul_eq_mul",
"pow_ne_zero",
"real.norm_of_nonneg",
"smul_unit_ball",
"support_comp_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
W_compact_support {D : ℝ} (Dpos : 0 < D) : has_compact_support (W D : E → ℝ) | begin
rw [has_compact_support_def, W_support E Dpos, closure_ball (0 : E) Dpos.ne'],
exact is_compact_closed_ball _ _,
end | lemma | exists_cont_diff_bump_base.W_compact_support | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"closure_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y (D : ℝ) : E → ℝ | W D ⋆[lsmul ℝ ℝ, μ] φ | def | exists_cont_diff_bump_base.Y | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | An auxiliary function to construct partitions of unity on finite-dimensional real vector spaces.
It is the convolution between a smooth function of integral `1` supported in the ball of radius `D`,
with the indicator function of the closed unit ball. Therefore, it is smooth, equal to `1` on the
ball of radius `1 - D`, ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Y_neg (D : ℝ) (x : E) : Y D (-x) = Y D x | begin
apply convolution_neg_of_neg_eq,
{ apply eventually_of_forall (λ x, _),
simp only [W_def, u_neg, smul_neg, algebra.id.smul_eq_mul, mul_eq_mul_left_iff,
eq_self_iff_true, true_or], },
{ apply eventually_of_forall (λ x, _),
simp only [φ, indicator, mem_closed_ball_zero_iff, norm_neg] },
end | lemma | exists_cont_diff_bump_base.Y_neg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"algebra.id.smul_eq_mul",
"convolution_neg_of_neg_eq",
"mul_eq_mul_left_iff",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_eq_one_of_mem_closed_ball {D : ℝ} {x : E} (Dpos : 0 < D)
(hx : x ∈ closed_ball (0 : E) (1 - D)) : Y D x = 1 | begin
change (W D ⋆[lsmul ℝ ℝ, μ] φ) x = 1,
have B : ∀ (y : E), y ∈ ball x D → φ y = 1,
{ have C : ball x D ⊆ ball 0 1,
{ apply ball_subset_ball',
simp only [mem_closed_ball] at hx,
linarith only [hx] },
assume y hy,
simp only [φ, indicator, mem_closed_ball, ite_eq_left_iff, not_le, zero_n... | lemma | exists_cont_diff_bump_base.Y_eq_one_of_mem_closed_ball | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"algebra.id.smul_eq_mul",
"convolution_eq_right'",
"ite_eq_left_iff",
"one_mul",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D)
(hx : x ∉ ball (0 : E) (1 + D)) : Y D x = 0 | begin
change (W D ⋆[lsmul ℝ ℝ, μ] φ) x = 0,
have B : ∀ y, y ∈ ball x D → φ y = 0,
{ assume y hy,
simp only [φ, indicator, mem_closed_ball_zero_iff, ite_eq_right_iff, one_ne_zero],
assume h'y,
have C : ball y D ⊆ ball 0 (1+D),
{ apply ball_subset_ball',
rw ← dist_zero_right at h'y,
lin... | lemma | exists_cont_diff_bump_base.Y_eq_zero_of_not_mem_ball | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"algebra.id.smul_eq_mul",
"convolution_eq_right'",
"ite_eq_right_iff",
"mul_zero",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_nonneg (D : ℝ) (x : E) : 0 ≤ Y D x | integral_nonneg (W_mul_φ_nonneg D x) | lemma | exists_cont_diff_bump_base.Y_nonneg | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_le_one {D : ℝ} (x : E) (Dpos : 0 < D) : Y D x ≤ 1 | begin
have A : (W D ⋆[lsmul ℝ ℝ, μ] φ) x ≤ (W D ⋆[lsmul ℝ ℝ, μ] 1) x,
{ apply convolution_mono_right_of_nonneg _ (W_nonneg D)
(indicator_le_self' (λ x hx, zero_le_one)) (λ x, zero_le_one),
refine (has_compact_support.convolution_exists_left _ (W_compact_support E Dpos) _
(locally_integrable_const (1... | lemma | exists_cont_diff_bump_base.Y_le_one | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"algebra.id.smul_eq_mul",
"continuous_linear_map.map_smul",
"convolution",
"convolution_mono_right_of_nonneg",
"has_compact_support.convolution_exists_left",
"mul_inv_rev",
"mul_one",
"pi.smul_apply",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < Y D x | begin
simp only [mem_ball_zero_iff] at hx,
refine (integral_pos_iff_support_of_nonneg (W_mul_φ_nonneg D x) _).2 _,
{ have F_comp : has_compact_support (W D),
from W_compact_support E Dpos,
have B : locally_integrable (φ : E → ℝ) μ,
from (locally_integrable_const _).indicator measurable_set_closed_... | lemma | exists_cont_diff_bump_base.Y_pos_of_mem_ball | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"abs_div",
"abs_of_nonneg",
"add_tsub_cancel_right",
"continuous",
"div_le_iff",
"div_nonpos_of_nonpos_of_nonneg",
"div_pos",
"exists_prop",
"has_compact_support.convolution_exists_left",
"measurable_set_closed_ball",
"mul_eq_zero",
"mul_one",
"norm_smul",
"not_forall",
"one_div",
"one... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_smooth : cont_diff_on ℝ ⊤ (uncurry Y) ((Ioo (0 : ℝ) 1) ×ˢ (univ : set E)) | begin
have hs : is_open (Ioo (0 : ℝ) (1 : ℝ)), from is_open_Ioo,
have hk : is_compact (closed_ball (0 : E) 1), from proper_space.is_compact_closed_ball _ _,
refine cont_diff_on_convolution_left_with_param (lsmul ℝ ℝ) hs hk _ _ _,
{ rintros p x hp hx,
simp only [W, mul_inv_rev, algebra.id.smul_eq_mul, mul_eq... | lemma | exists_cont_diff_bump_base.Y_smooth | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [
"abs_inv",
"abs_of_nonneg",
"abs_pos_of_pos",
"algebra.id.smul_eq_mul",
"cont_diff_on",
"cont_diff_on.mul",
"cont_diff_on.norm",
"cont_diff_on.pow",
"cont_diff_on.smul",
"cont_diff_on_convolution_left_with_param",
"cont_diff_on_fst",
"cont_diff_on_snd",
"div_eq_inv_mul",
"div_lt_one",
"i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_support {D : ℝ} (Dpos : 0 < D) (D_lt_one : D < 1) :
support (Y D : E → ℝ) = ball (0 : E) (1 + D) | support_eq_iff.2 ⟨λ x hx, (Y_pos_of_mem_ball Dpos D_lt_one hx).ne',
λ x hx, Y_eq_zero_of_not_mem_ball Dpos hx⟩ | lemma | exists_cont_diff_bump_base.Y_support | analysis.calculus | src/analysis/calculus/bump_function_findim.lean | [
"analysis.calculus.series",
"analysis.convolution",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.measure.haar.normed_space",
"data.set.pointwise.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_neg_inv_glue (x : ℝ) : ℝ | if x ≤ 0 then 0 else exp (-x⁻¹) | def | exp_neg_inv_glue | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp"
] | `exp_neg_inv_glue` is the real function given by `x ↦ exp (-1/x)` for `x > 0` and `0`
for `x ≤ 0`. It is a basic building block to construct smooth partitions of unity. Its main property
is that it vanishes for `x ≤ 0`, it is positive for `x > 0`, and the junction between the two
behaviors is flat enough to retain smoo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P_aux : ℕ → ℝ[X] | | 0 := 1
| (n+1) := X^2 * (P_aux n).derivative + (1 - C ↑(2 * n) * X) * (P_aux n) | def | exp_neg_inv_glue.P_aux | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [] | Our goal is to prove that `exp_neg_inv_glue` is `C^∞`. For this, we compute its successive
derivatives for `x > 0`. The `n`-th derivative is of the form `P_aux n (x) exp(-1/x) / x^(2 n)`,
where `P_aux n` is computed inductively. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux (n : ℕ) (x : ℝ) : ℝ | if x ≤ 0 then 0 else (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n) | def | exp_neg_inv_glue.f_aux | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp"
] | Formula for the `n`-th derivative of `exp_neg_inv_glue`, as an auxiliary function `f_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_zero_eq : f_aux 0 = exp_neg_inv_glue | begin
ext x,
by_cases h : x ≤ 0,
{ simp [exp_neg_inv_glue, f_aux, h] },
{ simp [h, exp_neg_inv_glue, f_aux, ne_of_gt (not_le.1 h), P_aux] }
end | lemma | exp_neg_inv_glue.f_aux_zero_eq | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue"
] | The `0`-th auxiliary function `f_aux 0` coincides with `exp_neg_inv_glue`, by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_deriv (n : ℕ) (x : ℝ) (hx : x ≠ 0) :
has_deriv_at (λx, (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n))
((P_aux (n+1)).eval x * exp (-x⁻¹) / x^(2 * (n + 1))) x | begin
simp only [P_aux, eval_add, eval_sub, eval_mul, eval_pow, eval_X, eval_C, eval_one],
convert (((P_aux n).has_deriv_at x).mul
(((has_deriv_at_exp _).comp x (has_deriv_at_inv hx).neg))).div
(has_deriv_at_pow (2 * n) x) (pow_ne_zero _ hx) using 1,
rw div_eq_div_iff,
{ have := pow_n... | lemma | exp_neg_inv_glue.f_aux_deriv | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"div_eq_div_iff",
"exp",
"has_deriv_at",
"has_deriv_at_exp",
"has_deriv_at_inv",
"has_deriv_at_pow",
"mul_one",
"mul_zero",
"nat.cast_zero",
"pow_ne_zero",
"ring"
] | For positive values, the derivative of the `n`-th auxiliary function `f_aux n`
(given in this statement in unfolded form) is the `n+1`-th auxiliary function, since
the polynomial `P_aux (n+1)` was chosen precisely to ensure this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_deriv_pos (n : ℕ) (x : ℝ) (hx : 0 < x) :
has_deriv_at (f_aux n) ((P_aux (n+1)).eval x * exp (-x⁻¹) / x^(2 * (n + 1))) x | begin
apply (f_aux_deriv n x (ne_of_gt hx)).congr_of_eventually_eq,
filter_upwards [lt_mem_nhds hx] with _ hy,
simp [f_aux, hy.not_le]
end | lemma | exp_neg_inv_glue.f_aux_deriv_pos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp",
"has_deriv_at",
"lt_mem_nhds"
] | For positive values, the derivative of the `n`-th auxiliary function `f_aux n`
is the `n+1`-th auxiliary function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_limit (n : ℕ) :
tendsto (λx, (P_aux n).eval x * exp (-x⁻¹) / x^(2 * n)) (𝓝[>] 0) (𝓝 0) | begin
have A : tendsto (λx, (P_aux n).eval x) (𝓝[>] 0) (𝓝 ((P_aux n).eval 0)) :=
(P_aux n).continuous_within_at,
have B : tendsto (λx, exp (-x⁻¹) / x^(2 * n)) (𝓝[>] 0) (𝓝 0),
{ convert (tendsto_pow_mul_exp_neg_at_top_nhds_0 (2 * n)).comp tendsto_inv_zero_at_top,
ext x,
field_simp },
convert A.mul ... | lemma | exp_neg_inv_glue.f_aux_limit | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous_within_at",
"exp",
"mul_div_assoc",
"tendsto_inv_zero_at_top"
] | To get differentiability at `0` of the auxiliary functions, we need to know that their limit
is `0`, to be able to apply general differentiability extension theorems. This limit is checked in
this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_deriv_zero (n : ℕ) : has_deriv_at (f_aux n) 0 0 | begin
-- we check separately differentiability on the left and on the right
have A : has_deriv_within_at (f_aux n) (0 : ℝ) (Iic 0) 0,
{ apply (has_deriv_at_const (0 : ℝ) (0 : ℝ)).has_deriv_within_at.congr,
{ assume y hy,
simp at hy,
simp [f_aux, hy] },
{ simp [f_aux, le_refl] } },
have B : h... | lemma | exp_neg_inv_glue.f_aux_deriv_zero | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"continuous_within_at",
"deriv",
"differentiable_at.differentiable_within_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at_const",
"has_deriv_at_interval_left_endpoint_of_tendsto_deriv",
"has_deriv_within_at",
"has_deriv_within_at.congr",
"self_mem_nhds_within"
] | Deduce from the limiting behavior at `0` of its derivative and general differentiability
extension theorems that the auxiliary function `f_aux n` is differentiable at `0`,
with derivative `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_has_deriv_at (n : ℕ) (x : ℝ) : has_deriv_at (f_aux n) (f_aux (n+1) x) x | begin
-- check separately the result for `x < 0`, where it is trivial, for `x > 0`, where it is done
-- in `f_aux_deriv_pos`, and for `x = 0`, done in
-- `f_aux_deriv_zero`.
rcases lt_trichotomy x 0 with hx|hx|hx,
{ have : f_aux (n+1) x = 0, by simp [f_aux, le_of_lt hx],
rw this,
apply (has_deriv_at_c... | lemma | exp_neg_inv_glue.f_aux_has_deriv_at | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp",
"gt_mem_nhds",
"has_deriv_at",
"has_deriv_at_const"
] | At every point, the auxiliary function `f_aux n` has a derivative which is
equal to `f_aux (n+1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_aux_iterated_deriv (n : ℕ) : iterated_deriv n (f_aux 0) = f_aux n | begin
induction n with n IH,
{ simp },
{ simp [iterated_deriv_succ, IH],
ext x,
exact (f_aux_has_deriv_at n x).deriv }
end | lemma | exp_neg_inv_glue.f_aux_iterated_deriv | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"deriv",
"iterated_deriv",
"iterated_deriv_succ"
] | The successive derivatives of the auxiliary function `f_aux 0` are the
functions `f_aux n`, by induction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff {n} : cont_diff ℝ n exp_neg_inv_glue | begin
rw ← f_aux_zero_eq,
apply cont_diff_of_differentiable_iterated_deriv (λ m hm, _),
rw f_aux_iterated_deriv m,
exact λ x, (f_aux_has_deriv_at m x).differentiable_at
end | theorem | exp_neg_inv_glue.cont_diff | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"cont_diff",
"cont_diff_of_differentiable_iterated_deriv",
"differentiable_at",
"exp_neg_inv_glue"
] | The function `exp_neg_inv_glue` is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_of_nonpos {x : ℝ} (hx : x ≤ 0) : exp_neg_inv_glue x = 0 | by simp [exp_neg_inv_glue, hx] | lemma | exp_neg_inv_glue.zero_of_nonpos | analysis.calculus | src/analysis/calculus/bump_function_inner.lean | [
"analysis.calculus.deriv.inv",
"analysis.calculus.extend_deriv",
"analysis.calculus.iterated_deriv",
"analysis.inner_product_space.calculus",
"analysis.special_functions.exp_deriv",
"measure_theory.integral.set_integral"
] | [
"exp_neg_inv_glue"
] | The function `exp_neg_inv_glue` vanishes on `(-∞, 0]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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