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 : set (ι → ℝ) | ⋃ J ∈ π, ↑J | def | box_integral.prepartition.Union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | Given a prepartition `π : box_integral.prepartition I`, `π.Union` is the part of `I` covered by
the boxes of `π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Union_def : π.Union = ⋃ J ∈ π, ↑J | rfl | lemma | box_integral.prepartition.Union_def | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_def' : π.Union = ⋃ J ∈ π.boxes, ↑J | rfl | lemma | box_integral.prepartition.Union_def' | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Union : x ∈ π.Union ↔ ∃ J ∈ π, x ∈ J | set.mem_Union₂ | lemma | box_integral.prepartition.mem_Union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.mem_Union₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_single (h : J ≤ I) : (single I J h).Union = J | by simp [Union_def] | lemma | box_integral.prepartition.Union_single | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_top : (⊤ : prepartition I).Union = I | by simp [prepartition.Union] | lemma | box_integral.prepartition.Union_top | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_eq_empty : π₁.Union = ∅ ↔ π₁ = ⊥ | by simp [← injective_boxes.eq_iff, finset.ext_iff, prepartition.Union, imp_false] | lemma | box_integral.prepartition.Union_eq_empty | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.ext_iff",
"imp_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_bot : (⊥ : prepartition I).Union = ∅ | Union_eq_empty.2 rfl | lemma | box_integral.prepartition.Union_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_Union (h : J ∈ π) : ↑J ⊆ π.Union | subset_bUnion_of_mem h | lemma | box_integral.prepartition.subset_Union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_subset : π.Union ⊆ I | Union₂_subset π.le_of_mem' | lemma | box_integral.prepartition.Union_subset | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_mono (h : π₁ ≤ π₂) : π₁.Union ⊆ π₂.Union | λ x hx, let ⟨J₁, hJ₁, hx⟩ := π₁.mem_Union.1 hx, ⟨J₂, hJ₂, hle⟩ := h hJ₁
in π₂.mem_Union.2 ⟨J₂, hJ₂, hle hx⟩ | lemma | box_integral.prepartition.Union_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_boxes_of_disjoint_Union (h : disjoint π₁.Union π₂.Union) :
disjoint π₁.boxes π₂.boxes | finset.disjoint_left.2 $ λ J h₁ h₂,
disjoint.le_bot (h.mono (π₁.subset_Union h₁) (π₂.subset_Union h₂)) ⟨J.upper_mem, J.upper_mem⟩ | lemma | box_integral.prepartition.disjoint_boxes_of_disjoint_Union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"disjoint.le_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_nonempty_imp_le_and_Union_subset : π₁ ≤ π₂ ↔
(∀ (J ∈ π₁) (J' ∈ π₂), (J ∩ J' : set (ι → ℝ)).nonempty → J ≤ J') ∧ π₁.Union ⊆ π₂.Union | begin
fsplit,
{ refine λ H, ⟨λ J hJ J' hJ' Hne, _, Union_mono H⟩,
rcases H hJ with ⟨J'', hJ'', Hle⟩, rcases Hne with ⟨x, hx, hx'⟩,
rwa π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx) },
{ rintro ⟨H, HU⟩ J hJ, simp only [set.subset_def, mem_Union] at HU,
rcases HU J.upper ⟨J, hJ, J.upper_mem⟩ with ⟨J₂, hJ₂, ... | lemma | box_integral.prepartition.le_iff_nonempty_imp_le_and_Union_subset | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_boxes_subset_Union_superset (h₁ : π₁.boxes ⊆ π₂.boxes) (h₂ : π₂.Union ⊆ π₁.Union) :
π₁ = π₂ | le_antisymm (λ J hJ, ⟨J, h₁ hJ, le_rfl⟩) $ le_iff_nonempty_imp_le_and_Union_subset.2
⟨λ J₁ hJ₁ J₂ hJ₂ Hne, (π₂.eq_of_mem_of_mem hJ₁ (h₁ hJ₂) Hne.some_spec.1 Hne.some_spec.2).le, h₂⟩ | lemma | box_integral.prepartition.eq_of_boxes_subset_Union_superset | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion (πi : Π J : box ι, prepartition J) : prepartition I | { boxes := π.boxes.bUnion $ λ J, (πi J).boxes,
le_of_mem' := λ J hJ,
begin
simp only [finset.mem_bUnion, exists_prop, mem_boxes] at hJ,
rcases hJ with ⟨J', hJ', hJ⟩,
exact ((πi J').le_of_mem hJ).trans (π.le_of_mem hJ')
end,
pairwise_disjoint :=
begin
simp only [set.pairwise, fins... | def | box_integral.prepartition.bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"exists_prop",
"finset.mem_bUnion",
"finset.mem_coe",
"set.disjoint_left",
"set.pairwise"
] | Given a prepartition `π` of a box `I` and a collection of prepartitions `πi J` of all boxes
`J ∈ π`, returns the prepartition of `I` into the union of the boxes of all `πi J`.
Though we only use the values of `πi` on the boxes of `π`, we require `πi` to be a globally defined
function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_bUnion : J ∈ π.bUnion πi ↔ ∃ J' ∈ π, J ∈ πi J' | by simp [bUnion] | lemma | box_integral.prepartition.mem_bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_le (πi : Π J, prepartition J) : π.bUnion πi ≤ π | λ J hJ, let ⟨J', hJ', hJ⟩ := π.mem_bUnion.1 hJ in ⟨J', hJ', (πi J').le_of_mem hJ⟩ | lemma | box_integral.prepartition.bUnion_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_top : π.bUnion (λ _, ⊤) = π | by { ext, simp } | lemma | box_integral.prepartition.bUnion_top | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_congr (h : π₁ = π₂) (hi : ∀ J ∈ π₁, πi₁ J = πi₂ J) :
π₁.bUnion πi₁ = π₂.bUnion πi₂ | by { subst π₂, ext J, simp [hi] { contextual := tt } } | lemma | box_integral.prepartition.bUnion_congr | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_congr_of_le (h : π₁ = π₂) (hi : ∀ J ≤ I, πi₁ J = πi₂ J) :
π₁.bUnion πi₁ = π₂.bUnion πi₂ | bUnion_congr h $ λ J hJ, hi J (π₁.le_of_mem hJ) | lemma | box_integral.prepartition.bUnion_congr_of_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_bUnion (πi : Π J : box ι, prepartition J) :
(π.bUnion πi).Union = ⋃ J ∈ π, (πi J).Union | by simp [prepartition.Union] | lemma | box_integral.prepartition.Union_bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_bUnion_boxes {M : Type*} [add_comm_monoid M] (π : prepartition I)
(πi : Π J, prepartition J) (f : box ι → M) :
∑ J in π.boxes.bUnion (λ J, (πi J).boxes), f J = ∑ J in π.boxes, ∑ J' in (πi J).boxes, f J' | begin
refine finset.sum_bUnion (λ J₁ h₁ J₂ h₂ hne, finset.disjoint_left.2 $ λ J' h₁' h₂', _),
exact hne (π.eq_of_le_of_le h₁ h₂ ((πi J₁).le_of_mem h₁') ((πi J₂).le_of_mem h₂'))
end | lemma | box_integral.prepartition.sum_bUnion_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_index (πi : Π J, prepartition J) (J : box ι) :
box ι | if hJ : J ∈ π.bUnion πi then (π.mem_bUnion.1 hJ).some else I | def | box_integral.prepartition.bUnion_index | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | Given a box `J ∈ π.bUnion πi`, returns the box `J' ∈ π` such that `J ∈ πi J'`.
For `J ∉ π.bUnion πi`, returns `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bUnion_index_mem (hJ : J ∈ π.bUnion πi) :
π.bUnion_index πi J ∈ π | by { rw [bUnion_index, dif_pos hJ], exact (π.mem_bUnion.1 hJ).some_spec.fst } | lemma | box_integral.prepartition.bUnion_index_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_index_le (πi : Π J, prepartition J) (J : box ι) : π.bUnion_index πi J ≤ I | begin
by_cases hJ : J ∈ π.bUnion πi,
{ exact π.le_of_mem (π.bUnion_index_mem hJ) },
{ rw [bUnion_index, dif_neg hJ], exact le_rfl }
end | lemma | box_integral.prepartition.bUnion_index_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bUnion_index (hJ : J ∈ π.bUnion πi) : J ∈ πi (π.bUnion_index πi J) | by convert (π.mem_bUnion.1 hJ).some_spec.snd; exact dif_pos hJ | lemma | box_integral.prepartition.mem_bUnion_index | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_bUnion_index (hJ : J ∈ π.bUnion πi) : J ≤ π.bUnion_index πi J | le_of_mem _ (π.mem_bUnion_index hJ) | lemma | box_integral.prepartition.le_bUnion_index | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_index_of_mem (hJ : J ∈ π) {J'} (hJ' : J' ∈ πi J) : π.bUnion_index πi J' = J | have J' ∈ π.bUnion πi, from π.mem_bUnion.2 ⟨J, hJ, hJ'⟩,
π.eq_of_le_of_le (π.bUnion_index_mem this) hJ (π.le_bUnion_index this) (le_of_mem _ hJ') | lemma | box_integral.prepartition.bUnion_index_of_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | Uniqueness property of `box_integral.partition.bUnion_index`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bUnion_assoc (πi : Π J, prepartition J) (πi' : box ι → Π J : box ι, prepartition J) :
π.bUnion (λ J, (πi J).bUnion (πi' J)) = (π.bUnion πi).bUnion (λ J, πi' (π.bUnion_index πi J) J) | begin
ext J,
simp only [mem_bUnion, exists_prop],
fsplit,
{ rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩,
refine ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, _⟩,
rwa π.bUnion_index_of_mem hJ₁ hJ₂ },
{ rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩,
refine ⟨J₂, hJ₂, J₁, hJ₁, _⟩,
rwa π.bUnion_index_of_mem hJ₂ hJ₁ at hJ }
end | lemma | box_integral.prepartition.bUnion_assoc | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_with_bot (boxes : finset (with_bot (box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : with_bot (box ι)) ≤ I)
(pairwise_disjoint : set.pairwise (boxes : set (with_bot (box ι))) disjoint) :
prepartition I | { boxes := boxes.erase_none,
le_of_mem' := λ J hJ,
begin
rw mem_erase_none at hJ,
simpa only [with_bot.some_eq_coe, with_bot.coe_le_coe] using le_of_mem _ hJ
end,
pairwise_disjoint := λ J₁ h₁ J₂ h₂ hne,
begin
simp only [mem_coe, mem_erase_none] at h₁ h₂,
exact box.disjoint_coe.1 ... | def | box_integral.prepartition.of_with_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"set.pairwise",
"with_bot",
"with_bot.coe_le_coe",
"with_bot.some_eq_coe"
] | Create a `box_integral.prepartition` from a collection of possibly empty boxes by filtering out
the empty one if it exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_with_bot {boxes : finset (with_bot (box ι))} {h₁ h₂} :
J ∈ (of_with_bot boxes h₁ h₂ : prepartition I) ↔ (J : with_bot (box ι)) ∈ boxes | mem_erase_none | lemma | box_integral.prepartition.mem_of_with_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_of_with_bot (boxes : finset (with_bot (box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : with_bot (box ι)) ≤ I)
(pairwise_disjoint : set.pairwise (boxes : set (with_bot (box ι))) disjoint) :
(of_with_bot boxes le_of_mem pairwise_disjoint).Union = ⋃ J ∈ boxes, ↑J | begin
suffices : (⋃ (J : box ι) (hJ : ↑J ∈ boxes), ↑J) = ⋃ J ∈ boxes, ↑J,
by simpa [of_with_bot, prepartition.Union],
simp only [← box.bUnion_coe_eq_coe, @Union_comm _ _ (box ι), @Union_comm _ _ (@eq _ _ _),
Union_Union_eq_right]
end | lemma | box_integral.prepartition.Union_of_with_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"set.pairwise",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_with_bot_le {boxes : finset (with_bot (box ι))}
{le_of_mem : ∀ J ∈ boxes, (J : with_bot (box ι)) ≤ I}
{pairwise_disjoint : set.pairwise (boxes : set (with_bot (box ι))) disjoint}
(H : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J') :
of_with_bot boxes le_of_mem pairwise_disjoint ≤ π | have ∀ (J : box ι), ↑J ∈ boxes → ∃ J' ∈ π, J ≤ J',
from λ J hJ, by simpa only [with_bot.coe_le_coe] using H J hJ with_bot.coe_ne_bot,
by simpa [of_with_bot, le_def] | lemma | box_integral.prepartition.of_with_bot_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"set.pairwise",
"with_bot",
"with_bot.coe_le_coe",
"with_bot.coe_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_with_bot {boxes : finset (with_bot (box ι))}
{le_of_mem : ∀ J ∈ boxes, (J : with_bot (box ι)) ≤ I}
{pairwise_disjoint : set.pairwise (boxes : set (with_bot (box ι))) disjoint}
(H : ∀ J ∈ π, ∃ J' ∈ boxes, ↑J ≤ J') :
π ≤ of_with_bot boxes le_of_mem pairwise_disjoint | begin
intros J hJ,
rcases H J hJ with ⟨J', J'mem, hle⟩,
lift J' to box ι using ne_bot_of_le_ne_bot with_bot.coe_ne_bot hle,
exact ⟨J', mem_of_with_bot.2 J'mem, with_bot.coe_le_coe.1 hle⟩
end | lemma | box_integral.prepartition.le_of_with_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"lift",
"ne_bot_of_le_ne_bot",
"set.pairwise",
"with_bot",
"with_bot.coe_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_with_bot_mono {boxes₁ : finset (with_bot (box ι))}
{le_of_mem₁ : ∀ J ∈ boxes₁, (J : with_bot (box ι)) ≤ I}
{pairwise_disjoint₁ : set.pairwise (boxes₁ : set (with_bot (box ι))) disjoint}
{boxes₂ : finset (with_bot (box ι))}
{le_of_mem₂ : ∀ J ∈ boxes₂, (J : with_bot (box ι)) ≤ I}
{pairwise_disjoint₂ : set.pa... | le_of_with_bot _ $ λ J hJ, H J (mem_of_with_bot.1 hJ) with_bot.coe_ne_bot | lemma | box_integral.prepartition.of_with_bot_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint",
"finset",
"set.pairwise",
"with_bot",
"with_bot.coe_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_of_with_bot {M : Type*} [add_comm_monoid M]
(boxes : finset (with_bot (box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : with_bot (box ι)) ≤ I)
(pairwise_disjoint : set.pairwise (boxes : set (with_bot (box ι))) disjoint)
(f : box ι → M) :
∑ J in (of_with_bot boxes le_of_mem pairwise_disjoint).boxes, f J =
∑ J i... | finset.sum_erase_none _ _ | lemma | box_integral.prepartition.sum_of_with_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"add_comm_monoid",
"disjoint",
"finset",
"option.elim",
"set.pairwise",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (π : prepartition I) (J : box ι) :
prepartition J | of_with_bot (π.boxes.image (λ J', J ⊓ J'))
(λ J' hJ', by { rcases finset.mem_image.1 hJ' with ⟨J', -, rfl⟩, exact inf_le_left })
begin
simp only [set.pairwise, on_fun, finset.mem_coe, finset.mem_image],
rintro _ ⟨J₁, h₁, rfl⟩ _ ⟨J₂, h₂, rfl⟩ Hne,
have : J₁ ≠ J₂, by { rintro rfl, exact Hne rfl },
exa... | def | box_integral.prepartition.restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.mem_coe",
"finset.mem_image",
"inf_le_left",
"set.pairwise"
] | Restrict a prepartition to a box. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_restrict : J₁ ∈ π.restrict J ↔ ∃ (J' ∈ π), (J₁ : with_bot (box ι)) = J ⊓ J' | by simp [restrict, eq_comm] | lemma | box_integral.prepartition.mem_restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_restrict' : J₁ ∈ π.restrict J ↔ ∃ (J' ∈ π), (J₁ : set (ι → ℝ)) = J ∩ J' | by simp only [mem_restrict, ← box.with_bot_coe_inj, box.coe_inf, box.coe_coe] | lemma | box_integral.prepartition.mem_restrict' | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_mono {π₁ π₂ : prepartition I} (Hle : π₁ ≤ π₂) :
π₁.restrict J ≤ π₂.restrict J | begin
refine of_with_bot_mono (λ J₁ hJ₁ hne, _),
rw finset.mem_image at hJ₁, rcases hJ₁ with ⟨J₁, hJ₁, rfl⟩,
rcases Hle hJ₁ with ⟨J₂, hJ₂, hle⟩,
exact ⟨_, finset.mem_image_of_mem _ hJ₂, inf_le_inf_left _ $ with_bot.coe_le_coe.2 hle⟩
end | lemma | box_integral.prepartition.restrict_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.mem_image",
"finset.mem_image_of_mem",
"inf_le_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_restrict : monotone (λ π : prepartition I, restrict π J) | λ π₁ π₂, restrict_mono | lemma | box_integral.prepartition.monotone_restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_boxes_of_le (π : prepartition I) (h : I ≤ J) :
(π.restrict J).boxes = π.boxes | begin
simp only [restrict, of_with_bot, erase_none_eq_bUnion],
refine finset.image_bUnion.trans _,
refine (finset.bUnion_congr rfl _).trans finset.bUnion_singleton_eq_self,
intros J' hJ',
rw [inf_of_le_right, ← with_bot.some_eq_coe, option.to_finset_some],
exact with_bot.coe_le_coe.2 ((π.le_of_mem hJ').tran... | lemma | box_integral.prepartition.restrict_boxes_of_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.bUnion_congr",
"finset.bUnion_singleton_eq_self",
"option.to_finset_some",
"with_bot.some_eq_coe"
] | Restricting to a larger box does not change the set of boxes. We cannot claim equality
of prepartitions because they have different types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_self : π.restrict I = π | injective_boxes $ restrict_boxes_of_le π le_rfl | lemma | box_integral.prepartition.restrict_self | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_restrict : (π.restrict J).Union = J ∩ π.Union | by simp [restrict, ← inter_Union, ← Union_def] | lemma | box_integral.prepartition.Union_restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_bUnion (πi : Π J, prepartition J) (hJ : J ∈ π) :
(π.bUnion πi).restrict J = πi J | begin
refine (eq_of_boxes_subset_Union_superset (λ J₁ h₁, _) _).symm,
{ refine (mem_restrict _).2 ⟨J₁, π.mem_bUnion.2 ⟨J, hJ, h₁⟩, (inf_of_le_right _).symm⟩,
exact with_bot.coe_le_coe.2 (le_of_mem _ h₁) },
{ simp only [Union_restrict, Union_bUnion, set.subset_def, set.mem_inter_iff, set.mem_Union],
rintro... | lemma | box_integral.prepartition.restrict_bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.mem_Union",
"set.mem_inter_iff",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_le_iff {πi : Π J, prepartition J} {π' : prepartition I} :
π.bUnion πi ≤ π' ↔ ∀ J ∈ π, πi J ≤ π'.restrict J | begin
fsplit; intros H J hJ,
{ rw ← π.restrict_bUnion πi hJ, exact restrict_mono H },
{ rw mem_bUnion at hJ, rcases hJ with ⟨J₁, h₁, hJ⟩,
rcases H J₁ h₁ hJ with ⟨J₂, h₂, Hle⟩,
rcases π'.mem_restrict.mp h₂ with ⟨J₃, h₃, H⟩,
exact ⟨J₃, h₃, Hle.trans $ with_bot.coe_le_coe.1 $ H.trans_le inf_le_right⟩ }
e... | lemma | box_integral.prepartition.bUnion_le_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_bUnion_iff {πi : Π J, prepartition J} {π' : prepartition I} :
π' ≤ π.bUnion πi ↔ π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J | begin
refine ⟨λ H, ⟨H.trans (π.bUnion_le πi), λ J hJ, _⟩, _⟩,
{ rw ← π.restrict_bUnion πi hJ, exact restrict_mono H },
{ rintro ⟨H, Hi⟩ J' hJ',
rcases H hJ' with ⟨J, hJ, hle⟩,
have : J' ∈ π'.restrict J,
from π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right $ with_bot.coe_le_coe.2 hle).symm⟩,
rcases ... | lemma | box_integral.prepartition.le_bUnion_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_def (π₁ π₂ : prepartition I) :
π₁ ⊓ π₂ = π₁.bUnion (λ J, π₂.restrict J) | rfl | lemma | box_integral.prepartition.inf_def | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {π₁ π₂ : prepartition I} :
J ∈ π₁ ⊓ π₂ ↔ ∃ (J₁ ∈ π₁) (J₂ ∈ π₂), (J : with_bot (box ι)) = J₁ ⊓ J₂ | by simp only [inf_def, mem_bUnion, mem_restrict] | lemma | box_integral.prepartition.mem_inf | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_inf (π₁ π₂ : prepartition I) : (π₁ ⊓ π₂).Union = π₁.Union ∩ π₂.Union | by simp only [inf_def, Union_bUnion, Union_restrict, ← Union_inter, ← Union_def] | lemma | box_integral.prepartition.Union_inf | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter (π : prepartition I) (p : box ι → Prop) : prepartition I | { boxes := π.boxes.filter p,
le_of_mem' := λ J hJ, π.le_of_mem (mem_filter.1 hJ).1,
pairwise_disjoint := λ J₁ h₁ J₂ h₂, π.disjoint_coe_of_mem (mem_filter.1 h₁).1
(mem_filter.1 h₂).1 } | def | box_integral.prepartition.filter | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"filter"
] | The prepartition with boxes `{J ∈ π | p J}`. | 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.prepartition.mem_filter | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.mem_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_le (π : prepartition I) (p : box ι → Prop) : π.filter p ≤ π | λ J hJ, let ⟨hπ, hp⟩ := π.mem_filter.1 hJ in ⟨J, hπ, le_rfl⟩ | lemma | box_integral.prepartition.filter_le | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_of_true {p : box ι → Prop} (hp : ∀ J ∈ π, p J) : π.filter p = π | by { ext J, simpa using hp J } | lemma | box_integral.prepartition.filter_of_true | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter_true : π.filter (λ _, true) = π | π.filter_of_true (λ _ _, trivial) | lemma | box_integral.prepartition.filter_true | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_filter_not (π : prepartition I) (p : box ι → Prop) :
(π.filter (λ J, ¬p J)).Union = π.Union \ (π.filter p).Union | begin
simp only [prepartition.Union],
convert (@set.bUnion_diff_bUnion_eq _ (box ι) π.boxes (π.filter p).boxes coe _).symm,
{ ext J x, simp { contextual := tt } },
{ convert π.pairwise_disjoint, simp }
end | lemma | box_integral.prepartition.Union_filter_not | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.bUnion_diff_bUnion_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_fiberwise {α M} [add_comm_monoid M] (π : prepartition I) (f : box ι → α) (g : box ι → M) :
∑ y in π.boxes.image f, ∑ J in (π.filter (λ J, f J = y)).boxes, g J = ∑ J in π.boxes, g J | by convert sum_fiberwise_of_maps_to (λ _, finset.mem_image_of_mem f) g | lemma | box_integral.prepartition.sum_fiberwise | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"add_comm_monoid",
"finset.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disj_union (π₁ π₂ : prepartition I) (h : disjoint π₁.Union π₂.Union) :
prepartition I | { boxes := π₁.boxes ∪ π₂.boxes,
le_of_mem' := λ J hJ, (finset.mem_union.1 hJ).elim π₁.le_of_mem π₂.le_of_mem,
pairwise_disjoint :=
suffices ∀ (J₁ ∈ π₁) (J₂ ∈ π₂), J₁ ≠ J₂ → disjoint (J₁ : set (ι → ℝ)) J₂,
by simpa [pairwise_union_of_symmetric (symmetric_disjoint.comap _), pairwise_disjoint],
λ J₁ h₁ J... | def | box_integral.prepartition.disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint"
] | Union of two disjoint prepartitions. | 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.prepartition.mem_disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.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 | by simp [disj_union, prepartition.Union, Union_or, Union_union_distrib] | lemma | box_integral.prepartition.Union_disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_disj_union_boxes {M : Type*} [add_comm_monoid M]
(h : disjoint π₁.Union π₂.Union) (f : box ι → M) :
∑ J in π₁.boxes ∪ π₂.boxes, f J = ∑ J in π₁.boxes, f J + ∑ J in π₂.boxes, f J | sum_union $ disjoint_boxes_of_disjoint_Union h | lemma | box_integral.prepartition.sum_disj_union_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"add_comm_monoid",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion : ℝ≥0 | π.boxes.sup box.distortion | def | box_integral.prepartition.distortion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | The distortion of a prepartition is the maximum of the distortions of the boxes of this
prepartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distortion_le_of_mem (h : J ∈ π) : J.distortion ≤ π.distortion | le_sup h | lemma | box_integral.prepartition.distortion_le_of_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_le_iff {c : ℝ≥0} : π.distortion ≤ c ↔ ∀ J ∈ π, box.distortion J ≤ c | finset.sup_le_iff | lemma | box_integral.prepartition.distortion_le_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"finset.sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_bUnion (π : prepartition I) (πi : Π J, prepartition J) :
(π.bUnion πi).distortion = π.boxes.sup (λ J, (πi J).distortion) | sup_bUnion _ _ | lemma | box_integral.prepartition.distortion_bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_disj_union (h : disjoint π₁.Union π₂.Union) :
(π₁.disj_union π₂ h).distortion = max π₁.distortion π₂.distortion | sup_union | lemma | box_integral.prepartition.distortion_disj_union | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_of_const {c} (h₁ : π.boxes.nonempty) (h₂ : ∀ J ∈ π, box.distortion J = c) :
π.distortion = c | (sup_congr rfl h₂).trans (sup_const h₁ _) | lemma | box_integral.prepartition.distortion_of_const | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_top (I : box ι) : distortion (⊤ : prepartition I) = I.distortion | sup_singleton | lemma | box_integral.prepartition.distortion_top | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distortion_bot (I : box ι) : distortion (⊥ : prepartition I) = 0 | sup_empty | lemma | box_integral.prepartition.distortion_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition (π : prepartition I) | ∀ x ∈ I, ∃ J ∈ π, x ∈ J | def | box_integral.prepartition.is_partition | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | A prepartition `π` of `I` is a partition if the boxes of `π` cover the whole `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_partition_iff_Union_eq {π : prepartition I} : π.is_partition ↔ π.Union = I | by simp_rw [is_partition, set.subset.antisymm_iff, π.Union_subset, true_and, set.subset_def,
mem_Union, box.mem_coe] | lemma | box_integral.prepartition.is_partition_iff_Union_eq | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"set.subset.antisymm_iff",
"set.subset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_single_iff (h : J ≤ I) : is_partition (single I J h) ↔ J = I | by simp [is_partition_iff_Union_eq] | lemma | box_integral.prepartition.is_partition_single_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_top (I : box ι) : is_partition (⊤ : prepartition I) | λ x hx, ⟨I, mem_top.2 rfl, hx⟩ | lemma | box_integral.prepartition.is_partition_top | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_eq (h : π.is_partition) : π.Union = I | is_partition_iff_Union_eq.1 h | lemma | box_integral.prepartition.is_partition.Union_eq | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_subset (h : π.is_partition) (π₁ : prepartition I) : π₁.Union ⊆ π.Union | h.Union_eq.symm ▸ π₁.Union_subset | lemma | box_integral.prepartition.is_partition.Union_subset | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique (h : π.is_partition) (hx : x ∈ I) :
∃! J ∈ π, x ∈ J | begin
rcases h x hx with ⟨J, h, hx⟩,
exact exists_unique.intro2 J h hx (λ J' h' hx', π.eq_of_mem_of_mem h' h hx' hx),
end | lemma | box_integral.prepartition.is_partition.exists_unique | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"exists_unique.intro2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_boxes (h : π.is_partition) : π.boxes.nonempty | let ⟨J, hJ, _⟩ := h _ I.upper_mem in ⟨J, hJ⟩ | lemma | box_integral.prepartition.is_partition.nonempty_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_boxes_subset (h₁ : π₁.is_partition) (h₂ : π₁.boxes ⊆ π₂.boxes) : π₁ = π₂ | eq_of_boxes_subset_Union_superset h₂ $ h₁.Union_subset _ | lemma | box_integral.prepartition.is_partition.eq_of_boxes_subset | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff (h : π₂.is_partition) :
π₁ ≤ π₂ ↔ ∀ (J ∈ π₁) (J' ∈ π₂), (J ∩ J' : set (ι → ℝ)).nonempty → J ≤ J' | le_iff_nonempty_imp_le_and_Union_subset.trans $ and_iff_left $ h.Union_subset _ | lemma | box_integral.prepartition.is_partition.le_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion (h : is_partition π) (hi : ∀ J ∈ π, is_partition (πi J)) :
is_partition (π.bUnion πi) | λ x hx, let ⟨J, hJ, hxi⟩ := h x hx, ⟨Ji, hJi, hx⟩ := hi J hJ x hxi in
⟨Ji, π.mem_bUnion.2 ⟨J, hJ, hJi⟩, hx⟩ | lemma | box_integral.prepartition.is_partition.bUnion | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (h : is_partition π) (hJ : J ≤ I) : is_partition (π.restrict J) | is_partition_iff_Union_eq.2 $ by simp [h.Union_eq, hJ] | lemma | box_integral.prepartition.is_partition.restrict | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf (h₁ : is_partition π₁) (h₂ : is_partition π₂) :
is_partition (π₁ ⊓ π₂) | is_partition_iff_Union_eq.2 $ by simp [h₁.Union_eq, h₂.Union_eq] | lemma | box_integral.prepartition.is_partition.inf | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_bUnion_partition (h : ∀ J ∈ π, (πi J).is_partition) : (π.bUnion πi).Union = π.Union | (Union_bUnion _ _).trans $ Union_congr_of_surjective id surjective_id $ λ J,
Union_congr_of_surjective id surjective_id $ λ hJ, (h J hJ).Union_eq | lemma | box_integral.prepartition.Union_bUnion_partition | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_disj_union_of_eq_diff (h : π₂.Union = I \ π₁.Union) :
is_partition (π₁.disj_union π₂ $ h.symm ▸ disjoint_sdiff_self_right) | is_partition_iff_Union_eq.2 $ (Union_disj_union _).trans $ by simp [h, π₁.Union_subset] | lemma | box_integral.prepartition.is_partition_disj_union_of_eq_diff | analysis.box_integral.partition | src/analysis/box_integral/partition/basic.lean | [
"algebra.big_operators.option",
"analysis.box_integral.box.basic"
] | [
"disjoint_sdiff_self_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integration_params : Type | (bRiemann bHenstock bDistortion : bool) | structure | box_integral.integration_params | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | An `integration_params` is a structure holding 3 boolean values used to define a filter to be
used in the definition of a box-integrable function.
* `bRiemann`: the value `tt` means that the filter corresponds to a Riemann-style integral, i.e. in
the definition of integrability we require a constant upper estimate `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_prod : integration_params ≃ bool × boolᵒᵈ × boolᵒᵈ | { to_fun := λ l, ⟨l.1, order_dual.to_dual l.2, order_dual.to_dual l.3⟩,
inv_fun := λ l, ⟨l.1, order_dual.of_dual l.2.1, order_dual.of_dual l.2.2⟩,
left_inv := λ ⟨a, b, c⟩, rfl,
right_inv := λ ⟨a, b, c⟩, rfl } | def | box_integral.integration_params.equiv_prod | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"inv_fun",
"order_dual.of_dual",
"order_dual.to_dual"
] | Auxiliary equivalence with a product type used to lift an order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_prod : integration_params ≃o bool × boolᵒᵈ × boolᵒᵈ | ⟨equiv_prod, λ ⟨x, y, z⟩, iff.rfl⟩ | def | box_integral.integration_params.iso_prod | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | Auxiliary `order_iso` with a product type used to lift a `bounded_order` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Riemann : integration_params | { bRiemann := tt,
bHenstock := tt,
bDistortion := ff } | def | box_integral.integration_params.Riemann | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | The `box_integral.integration_params` corresponding to the Riemann integral. In the
corresponding filter, we require that the diameters of all boxes `J` of a tagged partition are
bounded from above by a constant upper estimate that may not depend on the geometry of `J`, and each
tag belongs to the corresponding closed ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Henstock : integration_params | ⟨ff, tt, ff⟩ | def | box_integral.integration_params.Henstock | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | The `box_integral.integration_params` corresponding to the Henstock-Kurzweil integral. In the
corresponding filter, we require that the tagged partition is subordinate to a (possibly,
discontinuous) positive function `r` and each tag belongs to the corresponding closed box. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
McShane : integration_params | ⟨ff, ff, ff⟩ | def | box_integral.integration_params.McShane | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | The `box_integral.integration_params` corresponding to the McShane integral. In the
corresponding filter, we require that the tagged partition is subordinate to a (possibly,
discontinuous) positive function `r`; the tags may be outside of the corresponding closed box
(but still inside the ambient closed box `I.Icc`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GP : integration_params | ⊥ | def | box_integral.integration_params.GP | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | The `box_integral.integration_params` corresponding to the generalized Perron integral. In the
corresponding filter, we require that the tagged partition is subordinate to a (possibly,
discontinuous) positive function `r` and each tag belongs to the corresponding closed box. We also
require an upper estimate on the dis... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Henstock_le_Riemann : Henstock ≤ Riemann | dec_trivial | lemma | box_integral.integration_params.Henstock_le_Riemann | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Henstock_le_McShane : Henstock ≤ McShane | dec_trivial | lemma | box_integral.integration_params.Henstock_le_McShane | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
GP_le : GP ≤ l | bot_le | lemma | box_integral.integration_params.GP_le | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_base_set (l : integration_params) (I : box ι) (c : ℝ≥0)
(r : (ι → ℝ) → Ioi (0 : ℝ)) (π : tagged_prepartition I) : Prop | (is_subordinate : π.is_subordinate r)
(is_Henstock : l.bHenstock → π.is_Henstock)
(distortion_le : l.bDistortion → π.distortion ≤ c)
(exists_compl : l.bDistortion → ∃ π' : prepartition I, π'.Union = I \ π.Union ∧ π'.distortion ≤ c) | structure | box_integral.integration_params.mem_base_set | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | The predicate corresponding to a base set of the filter defined by an
`integration_params`. It says that
* if `l.bHenstock`, then `π` is a Henstock prepartition, i.e. each tag belongs to the corresponding
closed box;
* `π` is subordinate to `r`;
* if `l.bDistortion`, then the distortion of each box in `π` is less th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
r_cond {ι : Type*} (l : integration_params) (r : (ι → ℝ) → Ioi (0 : ℝ)) : Prop | l.bRiemann → ∀ x, r x = r 0 | def | box_integral.integration_params.r_cond | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | A predicate saying that in case `l.bRiemann = tt`, the function `r` is a constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_filter_distortion (l : integration_params) (I : box ι) (c : ℝ≥0) :
filter (tagged_prepartition I) | ⨅ (r : (ι → ℝ) → Ioi (0 : ℝ)) (hr : l.r_cond r), 𝓟 {π | l.mem_base_set I c r π} | def | box_integral.integration_params.to_filter_distortion | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"filter"
] | A set `s : set (tagged_prepartition I)` belongs to `l.to_filter_distortion I c` if there exists
a function `r : ℝⁿ → (0, ∞)` (or a constant `r` if `l.bRiemann = tt`) such that `s` contains each
prepartition `π` such that `l.mem_base_set I c r π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_filter (l : integration_params) (I : box ι) :
filter (tagged_prepartition I) | ⨆ c : ℝ≥0, l.to_filter_distortion I c | def | box_integral.integration_params.to_filter | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"filter"
] | A set `s : set (tagged_prepartition I)` belongs to `l.to_filter I` if for any `c : ℝ≥0` there
exists a function `r : ℝⁿ → (0, ∞)` (or a constant `r` if `l.bRiemann = tt`) such that
`s` contains each prepartition `π` such that `l.mem_base_set I c r π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_filter_distortion_Union (l : integration_params) (I : box ι) (c : ℝ≥0)
(π₀ : prepartition I) | l.to_filter_distortion I c ⊓ 𝓟 {π | π.Union = π₀.Union} | def | box_integral.integration_params.to_filter_distortion_Union | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | A set `s : set (tagged_prepartition I)` belongs to `l.to_filter_distortion_Union I c π₀` if
there exists a function `r : ℝⁿ → (0, ∞)` (or a constant `r` if `l.bRiemann = tt`) such that `s`
contains each prepartition `π` such that `l.mem_base_set I c r π` and `π.Union = π₀.Union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_filter_Union (l : integration_params) (I : box ι) (π₀ : prepartition I) | ⨆ c : ℝ≥0, l.to_filter_distortion_Union I c π₀ | def | box_integral.integration_params.to_filter_Union | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [] | A set `s : set (tagged_prepartition I)` belongs to `l.to_filter_Union I π₀` if for any `c : ℝ≥0`
there exists a function `r : ℝⁿ → (0, ∞)` (or a constant `r` if `l.bRiemann = tt`) such that `s`
contains each prepartition `π` such that `l.mem_base_set I c r π` and `π.Union = π₀.Union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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