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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r_cond_of_bRiemann_eq_ff {ι} (l : integration_params) (hl : l.bRiemann = ff)
{r : (ι → ℝ) → Ioi (0 : ℝ)} :
l.r_cond r | by simp [r_cond, hl] | lemma | box_integral.integration_params.r_cond_of_bRiemann_eq_ff | 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 | |
to_filter_inf_Union_eq (l : integration_params) (I : box ι) (π₀ : prepartition I) :
l.to_filter I ⊓ 𝓟 {π | π.Union = π₀.Union} = l.to_filter_Union I π₀ | (supr_inf_principal _ _).symm | lemma | box_integral.integration_params.to_filter_inf_Union_eq | 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 | |
mem_base_set.mono' (I : box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : tagged_prepartition I}
(hr : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J)) (hπ : l₁.mem_base_set I c₁ r₁ π) :
l₂.mem_base_set I c₂ r₂ π | ⟨hπ.1.mono' hr, λ h₂, hπ.2 (le_iff_imp.1 h.2.1 h₂),
λ hD, (hπ.3 (le_iff_imp.1 h.2.2 hD)).trans hc,
λ hD, (hπ.4 (le_iff_imp.1 h.2.2 hD)).imp $ λ π hπ, ⟨hπ.1, hπ.2.trans hc⟩⟩ | lemma | box_integral.integration_params.mem_base_set.mono' | 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 | |
mem_base_set.mono (I : box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : tagged_prepartition I}
(hr : ∀ x ∈ I.Icc, r₁ x ≤ r₂ x) (hπ : l₁.mem_base_set I c₁ r₁ π) :
l₂.mem_base_set I c₂ r₂ π | hπ.mono' I h hc $ λ J hJ, hr _ $ π.tag_mem_Icc J | lemma | box_integral.integration_params.mem_base_set.mono | 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 | |
mem_base_set.exists_common_compl (h₁ : l.mem_base_set I c₁ r₁ π₁)
(h₂ : l.mem_base_set I c₂ r₂ π₂) (hU : π₁.Union = π₂.Union) :
∃ π : prepartition I, π.Union = I \ π₁.Union ∧
(l.bDistortion → π.distortion ≤ c₁) ∧ (l.bDistortion → π.distortion ≤ c₂) | begin
wlog hc : c₁ ≤ c₂,
{ simpa [hU, and_comm] using this h₂ h₁ hU.symm (le_of_not_le hc) },
by_cases hD : (l.bDistortion : Prop),
{ rcases h₁.4 hD with ⟨π, hπU, hπc⟩,
exact ⟨π, hπU, λ _, hπc, λ _, hπc.trans hc⟩ },
{ exact ⟨π₁.to_prepartition.compl, π₁.to_prepartition.Union_compl,
λ h, (hD h).elim,... | lemma | box_integral.integration_params.mem_base_set.exists_common_compl | 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 | |
mem_base_set.union_compl_to_subordinate (hπ₁ : l.mem_base_set I c r₁ π₁)
(hle : ∀ x ∈ I.Icc, r₂ x ≤ r₁ x) {π₂ : prepartition I}
(hU : π₂.Union = I \ π₁.Union) (hc : l.bDistortion → π₂.distortion ≤ c) :
l.mem_base_set I c r₁ (π₁.union_compl_to_subordinate π₂ hU r₂) | ⟨hπ₁.1.disj_union ((π₂.is_subordinate_to_subordinate r₂).mono hle) _,
λ h, ((hπ₁.2 h).disj_union (π₂.is_Henstock_to_subordinate _) _),
λ h, (distortion_union_compl_to_subordinate _ _ _ _).trans_le (max_le (hπ₁.3 h) (hc h)),
λ _, ⟨⊥, by simp⟩⟩ | lemma | box_integral.integration_params.mem_base_set.union_compl_to_subordinate | 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 | |
mem_base_set.filter (hπ : l.mem_base_set I c r π) (p : box ι → Prop) :
l.mem_base_set I c r (π.filter p) | begin
refine ⟨λ J hJ, hπ.1 J (π.mem_filter.1 hJ).1, λ hH J hJ, hπ.2 hH J (π.mem_filter.1 hJ).1,
λ hD, (distortion_filter_le _ _).trans (hπ.3 hD), λ hD, _⟩,
rcases hπ.4 hD with ⟨π₁, hπ₁U, hc⟩,
set π₂ := π.filter (λ J, ¬p J),
have : disjoint π₁.Union π₂.Union,
by simpa [π₂, hπ₁U] using disjoint_sdiff_self... | lemma | box_integral.integration_params.mem_base_set.filter | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"disjoint",
"finset.filter_subset",
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_tagged_mem_base_set {π : prepartition I} {πi : Π J, tagged_prepartition J}
(h : ∀ J ∈ π, l.mem_base_set J c r (πi J)) (hp : ∀ J ∈ π, (πi J).is_partition)
(hc : l.bDistortion → π.compl.distortion ≤ c) :
l.mem_base_set I c r (π.bUnion_tagged πi) | begin
refine ⟨tagged_prepartition.is_subordinate_bUnion_tagged.2 $ λ J hJ, (h J hJ).1,
λ hH, tagged_prepartition.is_Henstock_bUnion_tagged.2 $ λ J hJ, (h J hJ).2 hH,
λ hD, _, λ hD, _⟩,
{ rw [prepartition.distortion_bUnion_tagged, finset.sup_le_iff],
exact λ J hJ, (h J hJ).3 hD },
{ refine ⟨_, _, hc hD... | lemma | box_integral.integration_params.bUnion_tagged_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"
] | [
"finset.sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
r_cond.mono {ι : Type*} {r : (ι → ℝ) → Ioi (0 : ℝ)} (h : l₁ ≤ l₂)
(hr : l₂.r_cond r) : l₁.r_cond r | λ hR, hr (le_iff_imp.1 h.1 hR) | lemma | box_integral.integration_params.r_cond.mono | 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 | |
r_cond.min {ι : Type*} {r₁ r₂ : (ι → ℝ) → Ioi (0 : ℝ)} (h₁ : l.r_cond r₁) (h₂ : l.r_cond r₂) :
l.r_cond (λ x, min (r₁ x) (r₂ x)) | λ hR x, congr_arg2 min (h₁ hR x) (h₂ hR x) | lemma | box_integral.integration_params.r_cond.min | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"congr_arg2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_distortion_mono (I : box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) :
l₁.to_filter_distortion I c₁ ≤ l₂.to_filter_distortion I c₂ | infi_mono $ λ r, infi_mono' $ λ hr,
⟨hr.mono h, principal_mono.2 $ λ _, mem_base_set.mono I h hc (λ _ _, le_rfl)⟩ | lemma | box_integral.integration_params.to_filter_distortion_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"infi_mono",
"infi_mono'",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_mono (I : box ι) {l₁ l₂ : integration_params} (h : l₁ ≤ l₂) :
l₁.to_filter I ≤ l₂.to_filter I | supr_mono $ λ c, to_filter_distortion_mono I h le_rfl | lemma | box_integral.integration_params.to_filter_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"le_rfl",
"supr_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_Union_mono (I : box ι) {l₁ l₂ : integration_params} (h : l₁ ≤ l₂)
(π₀ : prepartition I) :
l₁.to_filter_Union I π₀ ≤ l₂.to_filter_Union I π₀ | supr_mono $ λ c, inf_le_inf_right _ $ to_filter_distortion_mono _ h le_rfl | lemma | box_integral.integration_params.to_filter_Union_mono | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"inf_le_inf_right",
"le_rfl",
"supr_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_Union_congr (I : box ι) (l : integration_params) {π₁ π₂ : prepartition I}
(h : π₁.Union = π₂.Union) : l.to_filter_Union I π₁ = l.to_filter_Union I π₂ | by simp only [to_filter_Union, to_filter_distortion_Union, h] | lemma | box_integral.integration_params.to_filter_Union_congr | 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 | |
has_basis_to_filter_distortion (l : integration_params) (I : box ι) (c : ℝ≥0) :
(l.to_filter_distortion I c).has_basis l.r_cond (λ r, {π | l.mem_base_set I c r π}) | has_basis_binfi_principal'
(λ r₁ hr₁ r₂ hr₂, ⟨_, hr₁.min hr₂,
λ _, mem_base_set.mono _ le_rfl le_rfl (λ x hx, min_le_left _ _),
λ _, mem_base_set.mono _ le_rfl le_rfl (λ x hx, min_le_right _ _)⟩)
⟨λ _, ⟨1, zero_lt_one⟩, λ _ _, rfl⟩ | lemma | box_integral.integration_params.has_basis_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"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_to_filter_distortion_Union (l : integration_params) (I : box ι) (c : ℝ≥0)
(π₀ : prepartition I) :
(l.to_filter_distortion_Union I c π₀).has_basis l.r_cond
(λ r, {π | l.mem_base_set I c r π ∧ π.Union = π₀.Union}) | (l.has_basis_to_filter_distortion I c).inf_principal _ | lemma | box_integral.integration_params.has_basis_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_to_filter_Union (l : integration_params) (I : box ι) (π₀ : prepartition I) :
(l.to_filter_Union I π₀).has_basis (λ r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ), ∀ c, l.r_cond (r c))
(λ r, {π | ∃ c, l.mem_base_set I c (r c) π ∧ π.Union = π₀.Union}) | have _ := λ c, l.has_basis_to_filter_distortion_Union I c π₀,
by simpa only [set_of_and, set_of_exists] using has_basis_supr this | lemma | box_integral.integration_params.has_basis_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_to_filter_Union_top (l : integration_params) (I : box ι) :
(l.to_filter_Union I ⊤).has_basis (λ r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ), ∀ c, l.r_cond (r c))
(λ r, {π | ∃ c, l.mem_base_set I c (r c) π ∧ π.is_partition}) | by simpa only [tagged_prepartition.is_partition_iff_Union_eq, prepartition.Union_top]
using l.has_basis_to_filter_Union I ⊤ | lemma | box_integral.integration_params.has_basis_to_filter_Union_top | 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 | |
has_basis_to_filter (l : integration_params) (I : box ι) :
(l.to_filter I).has_basis (λ r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ), ∀ c, l.r_cond (r c))
(λ r, {π | ∃ c, l.mem_base_set I c (r c) π}) | by simpa only [set_of_exists] using has_basis_supr (l.has_basis_to_filter_distortion I) | lemma | box_integral.integration_params.has_basis_to_filter | 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 | |
tendsto_embed_box_to_filter_Union_top (l : integration_params) (h : I ≤ J) :
tendsto (tagged_prepartition.embed_box I J h) (l.to_filter_Union I ⊤)
(l.to_filter_Union J (prepartition.single J I h)) | begin
simp only [to_filter_Union, tendsto_supr], intro c,
set π₀ := (prepartition.single J I h),
refine le_supr_of_le (max c π₀.compl.distortion) _,
refine ((l.has_basis_to_filter_distortion_Union I c ⊤).tendsto_iff
(l.has_basis_to_filter_distortion_Union J _ _)).2 (λ r hr, _),
refine ⟨r, hr, λ π hπ, _⟩,
... | lemma | box_integral.integration_params.tendsto_embed_box_to_filter_Union_top | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"le_supr_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_base_set_le_Union_eq (l : integration_params) (π₀ : prepartition I)
(hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
∃ π, l.mem_base_set I c r π ∧ π.to_prepartition ≤ π₀ ∧ π.Union = π₀.Union | begin
rcases π₀.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r
with ⟨π, hle, hH, hr, hd, hU⟩,
refine ⟨π, ⟨hr, λ _, hH, λ _, hd.trans_le hc₁, λ hD, ⟨π₀.compl, _, hc₂⟩⟩, ⟨hle, hU⟩⟩,
exact prepartition.compl_congr hU ▸ π.to_prepartition.Union_compl
end | lemma | box_integral.integration_params.exists_mem_base_set_le_Union_eq | 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 | |
exists_mem_base_set_is_partition (l : integration_params) (I : box ι)
(hc : I.distortion ≤ c) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
∃ π, l.mem_base_set I c r π ∧ π.is_partition | begin
rw ← prepartition.distortion_top at hc,
have hc' : (⊤ : prepartition I).compl.distortion ≤ c, by simp,
simpa [is_partition_iff_Union_eq] using l.exists_mem_base_set_le_Union_eq ⊤ hc hc' r
end | lemma | box_integral.integration_params.exists_mem_base_set_is_partition | 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 | |
to_filter_distortion_Union_ne_bot (l : integration_params) (I : box ι)
(π₀ : prepartition I) (hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) :
(l.to_filter_distortion_Union I c π₀).ne_bot | ((l.has_basis_to_filter_distortion I _).inf_principal _).ne_bot_iff.2 $ λ r hr,
(l.exists_mem_base_set_le_Union_eq π₀ hc₁ hc₂ r).imp $ λ π hπ, ⟨hπ.1, hπ.2.2⟩ | lemma | box_integral.integration_params.to_filter_distortion_Union_ne_bot | 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 | |
to_filter_distortion_Union_ne_bot' (l : integration_params) (I : box ι)
(π₀ : prepartition I) :
(l.to_filter_distortion_Union I (max π₀.distortion π₀.compl.distortion) π₀).ne_bot | l.to_filter_distortion_Union_ne_bot I π₀ (le_max_left _ _) (le_max_right _ _) | instance | box_integral.integration_params.to_filter_distortion_Union_ne_bot' | 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 | |
to_filter_distortion_ne_bot (l : integration_params) (I : box ι) :
(l.to_filter_distortion I I.distortion).ne_bot | by simpa using (l.to_filter_distortion_Union_ne_bot' I ⊤).mono inf_le_left | instance | box_integral.integration_params.to_filter_distortion_ne_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_ne_bot (l : integration_params) (I : box ι) : (l.to_filter I).ne_bot | (l.to_filter_distortion_ne_bot I).mono $ le_supr _ _ | instance | box_integral.integration_params.to_filter_ne_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"le_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_filter_Union_ne_bot (l : integration_params) (I : box ι) (π₀ : prepartition I) :
(l.to_filter_Union I π₀).ne_bot | (l.to_filter_distortion_Union_ne_bot' I π₀).mono $
le_supr (λ c, l.to_filter_distortion_Union I c π₀) _ | instance | box_integral.integration_params.to_filter_Union_ne_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/filter.lean | [
"analysis.box_integral.partition.subbox_induction",
"analysis.box_integral.partition.split"
] | [
"le_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_is_partition (l : integration_params) (I : box ι) :
∀ᶠ π in l.to_filter_Union I ⊤, tagged_prepartition.is_partition π | eventually_supr.2 $ λ c, eventually_inf_principal.2 $ eventually_of_forall $
λ π h, π.is_partition_iff_Union_eq.2 (h.trans prepartition.Union_top) | lemma | box_integral.integration_params.eventually_is_partition | 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 | |
measure_Icc_lt_top (μ : measure (ι → ℝ)) [is_locally_finite_measure μ] : μ I.Icc < ∞ | show μ (Icc I.lower I.upper) < ∞, from I.is_compact_Icc.measure_lt_top | lemma | box_integral.box.measure_Icc_lt_top | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"measure_Icc_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_coe_lt_top (μ : measure (ι → ℝ)) [is_locally_finite_measure μ] : μ I < ∞ | (measure_mono $ coe_subset_Icc).trans_lt (I.measure_Icc_lt_top μ) | lemma | box_integral.box.measure_coe_lt_top | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_coe : measurable_set (I : set (ι → ℝ)) | by { rw coe_eq_pi, exact measurable_set.univ_pi (λ i, measurable_set_Ioc) } | lemma | box_integral.box.measurable_set_coe | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"measurable_set",
"measurable_set.univ_pi",
"measurable_set_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_Icc : measurable_set I.Icc | measurable_set_Icc | lemma | box_integral.box.measurable_set_Icc | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"measurable_set",
"measurable_set_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_Ioo : measurable_set I.Ioo | measurable_set.univ_pi $ λ i, measurable_set_Ioo | lemma | box_integral.box.measurable_set_Ioo | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"measurable_set",
"measurable_set.univ_pi",
"measurable_set_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ae_eq_Icc : (I : set (ι → ℝ)) =ᵐ[volume] I.Icc | by { rw coe_eq_pi, exact measure.univ_pi_Ioc_ae_eq_Icc } | lemma | box_integral.box.coe_ae_eq_Icc | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioo_ae_eq_Icc : I.Ioo =ᵐ[volume] I.Icc | measure.univ_pi_Ioo_ae_eq_Icc | lemma | box_integral.box.Ioo_ae_eq_Icc | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prepartition.measure_Union_to_real [finite ι] {I : box ι} (π : prepartition I)
(μ : measure (ι → ℝ)) [is_locally_finite_measure μ] :
(μ π.Union).to_real = ∑ J in π.boxes, (μ J).to_real | begin
erw [← ennreal.to_real_sum, π.Union_def, measure_bUnion_finset π.pairwise_disjoint],
exacts [λ J hJ, J.measurable_set_coe, λ J hJ, (J.measure_coe_lt_top μ).ne]
end | lemma | box_integral.prepartition.measure_Union_to_real | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"ennreal.to_real_sum",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_box_additive (μ : measure (ι → ℝ)) [is_locally_finite_measure μ] :
ι →ᵇᵃ[⊤] ℝ | { to_fun := λ J, (μ J).to_real,
sum_partition_boxes' := λ J hJ π hπ, by rw [← π.measure_Union_to_real, hπ.Union_eq] } | def | measure_theory.measure.to_box_additive | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [] | If `μ` is a locally finite measure on `ℝⁿ`, then `λ J, (μ J).to_real` is a box-additive
function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_apply (I : box ι) :
(volume : measure (ι → ℝ)).to_box_additive I = ∏ i, (I.upper i - I.lower i) | by rw [measure.to_box_additive_apply, coe_eq_pi, real.volume_pi_Ioc_to_real I.lower_le_upper] | lemma | box_integral.box.volume_apply | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"real.volume_pi_Ioc_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume_face_mul {n} (i : fin (n + 1)) (I : box (fin (n + 1))) :
(∏ j, ((I.face i).upper j - (I.face i).lower j)) * (I.upper i - I.lower i) =
∏ j, (I.upper j - I.lower j) | by simp only [face_lower, face_upper, (∘), fin.prod_univ_succ_above _ i, mul_comm] | lemma | box_integral.box.volume_face_mul | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"fin.prod_univ_succ_above",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
volume {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] :
ι →ᵇᵃ (E →L[ℝ] E) | (volume : measure (ι → ℝ)).to_box_additive.to_smul | def | box_integral.box_additive_map.volume | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"normed_add_comm_group",
"normed_space"
] | Box-additive map sending each box `I` to the continuous linear endomorphism
`x ↦ (volume I).to_real • x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_apply {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] (I : box ι) (x : E) :
box_additive_map.volume I x = (∏ j, (I.upper j - I.lower j)) • x | congr_arg2 (•) I.volume_apply rfl | lemma | box_integral.box_additive_map.volume_apply | analysis.box_integral.partition | src/analysis/box_integral/partition/measure.lean | [
"analysis.box_integral.partition.additive",
"measure_theory.measure.lebesgue.basic"
] | [
"congr_arg2",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower (I : box ι) (i : ι) (x : ℝ) : with_bot (box ι) | mk' I.lower (update I.upper i (min x (I.upper i))) | def | box_integral.box.split_lower | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"mk'",
"update",
"with_bot"
] | Given a box `I` and `x ∈ (I.lower i, I.upper i)`, the hyperplane `{y : ι → ℝ | y i = x}` splits
`I` into two boxes. `box_integral.box.split_lower I i x` is the box `I ∩ {y | y i ≤ x}`
(if it is nonempty). As usual, we represent a box that may be empty as
`with_bot (box_integral.box ι)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_split_lower : (split_lower I i x : set (ι → ℝ)) = I ∩ {y | y i ≤ x} | begin
rw [split_lower, coe_mk'],
ext y,
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_set_of_eq, forall_and_distrib,
← pi.le_def, le_update_iff, le_min_iff, and_assoc, and_forall_ne i, mem_def],
rw [and_comm (y i ≤ x), pi.le_def]
end | lemma | box_integral.box.coe_split_lower | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"and_forall_ne",
"forall_and_distrib",
"le_min_iff",
"le_update_iff",
"pi.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower_le : I.split_lower i x ≤ I | with_bot_coe_subset_iff.1 $ by simp | lemma | box_integral.box.split_lower_le | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower_eq_bot {i x} : I.split_lower i x = ⊥ ↔ x ≤ I.lower i | begin
rw [split_lower, mk'_eq_bot, exists_update_iff I.upper (λ j y, y ≤ I.lower j)],
simp [(I.lower_lt_upper _).not_le]
end | lemma | box_integral.box.split_lower_eq_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"exists_update_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower_eq_self : I.split_lower i x = I ↔ I.upper i ≤ x | by simp [split_lower, update_eq_iff] | lemma | box_integral.box.split_lower_eq_self | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"update_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower_def [decidable_eq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, I.lower j < update I.upper i x j :=
(forall_update_iff I.upper (λ j y, I.lower j < y)).2 ⟨h.1, λ j hne, I.lower_lt_upper _⟩) :
I.split_lower i x = (⟨I.lower, update I.upper i x, h'⟩ : box ι) | by { simp only [split_lower, mk'_eq_coe, min_eq_left h.2.le], use rfl, congr } | lemma | box_integral.box.split_lower_def | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"forall_update_iff",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_upper (I : box ι) (i : ι) (x : ℝ) : with_bot (box ι) | mk' (update I.lower i (max x (I.lower i))) I.upper | def | box_integral.box.split_upper | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"mk'",
"update",
"with_bot"
] | Given a box `I` and `x ∈ (I.lower i, I.upper i)`, the hyperplane `{y : ι → ℝ | y i = x}` splits
`I` into two boxes. `box_integral.box.split_upper I i x` is the box `I ∩ {y | x < y i}`
(if it is nonempty). As usual, we represent a box that may be empty as
`with_bot (box_integral.box ι)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_split_upper : (split_upper I i x : set (ι → ℝ)) = I ∩ {y | x < y i} | begin
rw [split_upper, coe_mk'],
ext y,
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_set_of_eq, forall_and_distrib,
forall_update_iff I.lower (λ j z, z < y j), max_lt_iff, and_assoc (x < y i),
and_forall_ne i, mem_def],
exact and_comm _ _
end | lemma | box_integral.box.coe_split_upper | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"and_forall_ne",
"forall_and_distrib",
"forall_update_iff",
"max_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_upper_le : I.split_upper i x ≤ I | with_bot_coe_subset_iff.1 $ by simp | lemma | box_integral.box.split_upper_le | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_upper_eq_bot {i x} : I.split_upper i x = ⊥ ↔ I.upper i ≤ x | begin
rw [split_upper, mk'_eq_bot, exists_update_iff I.lower (λ j y, I.upper j ≤ y)],
simp [(I.lower_lt_upper _).not_le]
end | lemma | box_integral.box.split_upper_eq_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"exists_update_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_upper_eq_self : I.split_upper i x = I ↔ x ≤ I.lower i | by simp [split_upper, update_eq_iff] | lemma | box_integral.box.split_upper_eq_self | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"update_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_upper_def [decidable_eq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, update I.lower i x j < I.upper j :=
(forall_update_iff I.lower (λ j y, y < I.upper j)).2 ⟨h.2, λ j hne, I.lower_lt_upper _⟩) :
I.split_upper i x = (⟨update I.lower i x, I.upper, h'⟩ : box ι) | by { simp only [split_upper, mk'_eq_coe, max_eq_left h.1.le], refine ⟨_, rfl⟩, congr } | lemma | box_integral.box.split_upper_def | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"forall_update_iff",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_split_lower_split_upper (I : box ι) (i : ι) (x : ℝ) :
disjoint (I.split_lower i x) (I.split_upper i x) | begin
rw [← disjoint_with_bot_coe, coe_split_lower, coe_split_upper],
refine (disjoint.inf_left' _ _).inf_right' _,
rw set.disjoint_left,
exact λ y (hle : y i ≤ x) hlt, not_lt_of_le hle hlt
end | lemma | box_integral.box.disjoint_split_lower_split_upper | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"disjoint.inf_left'",
"not_lt_of_le",
"set.disjoint_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_lower_ne_split_upper (I : box ι) (i : ι) (x : ℝ) :
I.split_lower i x ≠ I.split_upper i x | begin
cases le_or_lt x (I.lower i),
{ rw [split_upper_eq_self.2 h, split_lower_eq_bot.2 h], exact with_bot.bot_ne_coe },
{ refine (disjoint_split_lower_split_upper I i x).ne _,
rwa [ne.def, split_lower_eq_bot, not_le] }
end | lemma | box_integral.box.split_lower_ne_split_upper | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"with_bot.bot_ne_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split (I : box ι) (i : ι) (x : ℝ) : prepartition I | of_with_bot {I.split_lower i x, I.split_upper i x}
begin
simp only [finset.mem_insert, finset.mem_singleton],
rintro J (rfl|rfl),
exacts [box.split_lower_le, box.split_upper_le]
end
begin
simp only [finset.coe_insert, finset.coe_singleton, true_and, set.mem_singleton_iff,
pairwise_insert_of_... | def | box_integral.prepartition.split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset.coe_insert",
"finset.coe_singleton",
"finset.mem_insert",
"finset.mem_singleton",
"set.mem_singleton_iff",
"symmetric_disjoint"
] | The partition of `I : box ι` into the boxes `I ∩ {y | y ≤ x i}` and `I ∩ {y | x i < y}`.
One of these boxes can be empty, then this partition is just the single-box partition `⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_split_iff : J ∈ split I i x ↔ ↑J = I.split_lower i x ∨ ↑J = I.split_upper i x | by simp [split] | lemma | box_integral.prepartition.mem_split_iff | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_split_iff' : J ∈ split I i x ↔
(J : set (ι → ℝ)) = I ∩ {y | y i ≤ x} ∨ (J : set (ι → ℝ)) = I ∩ {y | x < y i} | by simp [mem_split_iff, ← box.with_bot_coe_inj] | lemma | box_integral.prepartition.mem_split_iff' | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_split (I : box ι) (i : ι) (x : ℝ) : (split I i x).Union = I | by simp [split, ← inter_union_distrib_left, ← set_of_or, le_or_lt] | lemma | box_integral.prepartition.Union_split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_split (I : box ι) (i : ι) (x : ℝ) : is_partition (split I i x) | is_partition_iff_Union_eq.2 $ Union_split I i x | lemma | box_integral.prepartition.is_partition_split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_split_boxes {M : Type*} [add_comm_monoid M] (I : box ι) (i : ι) (x : ℝ) (f : box ι → M) :
∑ J in (split I i x).boxes, f J = (I.split_lower i x).elim 0 f + (I.split_upper i x).elim 0 f | by rw [split, sum_of_with_bot, finset.sum_pair (I.split_lower_ne_split_upper i x)] | lemma | box_integral.prepartition.sum_split_boxes | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_of_not_mem_Ioo (h : x ∉ Ioo (I.lower i) (I.upper i)) : split I i x = ⊤ | begin
refine ((is_partition_top I).eq_of_boxes_subset (λ J hJ, _)).symm,
rcases mem_top.1 hJ with rfl, clear hJ,
rw [mem_boxes, mem_split_iff],
rw [mem_Ioo, not_and_distrib, not_lt, not_lt] at h,
cases h; [right, left],
{ rwa [eq_comm, box.split_upper_eq_self] },
{ rwa [eq_comm, box.split_lower_eq_self] }... | lemma | box_integral.prepartition.split_of_not_mem_Ioo | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"not_and_distrib"
] | If `x ∉ (I.lower i, I.upper i)`, then the hyperplane `{y | y i = x}` does not split `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_eq_of_mem_split_of_mem_le {y : ι → ℝ} (h₁ : J ∈ split I i x) (h₂ : y ∈ J) (h₃ : y i ≤ x) :
(J : set (ι → ℝ)) = I ∩ {y | y i ≤ x} | (mem_split_iff'.1 h₁).resolve_right $ λ H,
by { rw [← box.mem_coe, H] at h₂, exact h₃.not_lt h₂.2 } | lemma | box_integral.prepartition.coe_eq_of_mem_split_of_mem_le | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_of_mem_split_of_lt_mem {y : ι → ℝ} (h₁ : J ∈ split I i x) (h₂ : y ∈ J) (h₃ : x < y i) :
(J : set (ι → ℝ)) = I ∩ {y | x < y i} | (mem_split_iff'.1 h₁).resolve_left $ λ H,
by { rw [← box.mem_coe, H] at h₂, exact h₃.not_le h₂.2 } | lemma | box_integral.prepartition.coe_eq_of_mem_split_of_lt_mem | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_split (h : I ≤ J) (i : ι) (x : ℝ) : (split J i x).restrict I = split I i x | begin
refine ((is_partition_split J i x).restrict h).eq_of_boxes_subset _,
simp only [finset.subset_iff, mem_boxes, mem_restrict', exists_prop, mem_split_iff'],
have : ∀ s, (I ∩ s : set (ι → ℝ)) ⊆ J, from λ s, (inter_subset_left _ _).trans h,
rintro J₁ ⟨J₂, (H₂|H₂), H₁⟩; [left, right]; simp [H₁, H₂, inter_left_... | lemma | box_integral.prepartition.restrict_split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"exists_prop",
"finset.subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_split (π : prepartition I) (i : ι) (x : ℝ) :
π ⊓ split I i x = π.bUnion (λ J, split J i x) | bUnion_congr_of_le rfl $ λ J hJ, restrict_split hJ i x | lemma | box_integral.prepartition.inf_split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_many (I : box ι) (s : finset (ι × ℝ)) : prepartition I | s.inf (λ p, split I p.1 p.2) | def | box_integral.prepartition.split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset"
] | Split a box along many hyperplanes `{y | y i = x}`; each hyperplane is given by the pair
`(i x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_many_empty (I : box ι) : split_many I ∅ = ⊤ | finset.inf_empty | lemma | box_integral.prepartition.split_many_empty | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset.inf_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_many_insert (I : box ι) (s : finset (ι × ℝ)) (p : ι × ℝ) :
split_many I (insert p s) = split_many I s ⊓ split I p.1 p.2 | by rw [split_many, finset.inf_insert, inf_comm, split_many] | lemma | box_integral.prepartition.split_many_insert | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset",
"finset.inf_insert",
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_many_le_split (I : box ι) {s : finset (ι × ℝ)} {p : ι × ℝ} (hp : p ∈ s) :
split_many I s ≤ split I p.1 p.2 | finset.inf_le hp | lemma | box_integral.prepartition.split_many_le_split | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset",
"finset.inf_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition_split_many (I : box ι) (s : finset (ι × ℝ)) :
is_partition (split_many I s) | finset.induction_on s (by simp only [split_many_empty, is_partition_top]) $
λ a s ha hs, by simpa only [split_many_insert, inf_split]
using hs.bUnion (λ J hJ, is_partition_split _ _ _) | lemma | box_integral.prepartition.is_partition_split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset",
"finset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_split_many (I : box ι) (s : finset (ι × ℝ)) : (split_many I s).Union = I | (is_partition_split_many I s).Union_eq | lemma | box_integral.prepartition.Union_split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_split_many {I : box ι} (π : prepartition I) (s : finset (ι × ℝ)) :
π ⊓ split_many I s = π.bUnion (λ J, split_many J s) | begin
induction s using finset.induction_on with p s hp ihp,
{ simp },
{ simp_rw [split_many_insert, ← inf_assoc, ihp, inf_split, bUnion_assoc] }
end | lemma | box_integral.prepartition.inf_split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"finset",
"finset.induction_on",
"inf_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_disjoint_imp_le_of_subset_of_mem_split_many {I J Js : box ι} {s : finset (ι × ℝ)}
(H : ∀ i, {(i, J.lower i), (i, J.upper i)} ⊆ s) (HJs : Js ∈ split_many I s)
(Hn : ¬disjoint (J : with_bot (box ι)) Js) : Js ≤ J | begin
simp only [finset.insert_subset, finset.singleton_subset_iff] at H,
rcases box.not_disjoint_coe_iff_nonempty_inter.mp Hn with ⟨x, hx, hxs⟩,
refine λ y hy i, ⟨_, _⟩,
{ rcases split_many_le_split I (H i).1 HJs with ⟨Jl, Hmem : Jl ∈ split I i (J.lower i), Hle⟩,
have := Hle hxs,
rw [← box.coe_subset_c... | lemma | box_integral.prepartition.not_disjoint_imp_le_of_subset_of_mem_split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"finset",
"finset.insert_subset",
"finset.singleton_subset_iff",
"with_bot"
] | Let `s : finset (ι × ℝ)` be a set of hyperplanes `{x : ι → ℝ | x i = r}` in `ι → ℝ` encoded as
pairs `(i, r)`. Suppose that this set contains all faces of a box `J`. The hyperplanes of `s` split
a box `I` into subboxes. Let `Js` be one of them. If `J` and `Js` have nonempty intersection, then
`Js` is a subbox of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_not_disjoint_imp_le_of_mem_split_many (s : finset (box ι)) :
∀ᶠ t : finset (ι × ℝ) in at_top, ∀ (I : box ι) (J ∈ s) (J' ∈ split_many I t),
¬disjoint (J : with_bot (box ι)) J' → J' ≤ J | begin
casesI nonempty_fintype ι,
refine eventually_at_top.2
⟨s.bUnion (λ J, finset.univ.bUnion (λ i, {(i, J.lower i), (i, J.upper i)})),
λ t ht I J hJ J' hJ', not_disjoint_imp_le_of_subset_of_mem_split_many (λ i, _) hJ'⟩,
exact λ p hp, ht (finset.mem_bUnion.2 ⟨J, hJ, finset.mem_bUnion.2 ⟨i, finset.mem_u... | lemma | box_integral.prepartition.eventually_not_disjoint_imp_le_of_mem_split_many | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"disjoint",
"finset",
"finset.mem_univ",
"nonempty_fintype",
"with_bot"
] | Let `s` be a finite set of boxes in `ℝⁿ = ι → ℝ`. Then there exists a finite set `t₀` of
hyperplanes (namely, the set of all hyperfaces of boxes in `s`) such that for any `t ⊇ t₀`
and any box `I` in `ℝⁿ` the following holds. The hyperplanes from `t` split `I` into subboxes.
Let `J'` be one of them, and let `J` be one o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_split_many_inf_eq_filter (π : prepartition I) :
∀ᶠ t : finset (ι × ℝ) in at_top,
π ⊓ (split_many I t) = (split_many I t).filter (λ J, ↑J ⊆ π.Union) | begin
refine (eventually_not_disjoint_imp_le_of_mem_split_many π.boxes).mono (λ t ht, _),
refine le_antisymm ((bUnion_le_iff _).2 $ λ J hJ, _) (le_inf (λ J hJ, _) (filter_le _ _)),
{ refine of_with_bot_mono _,
simp only [finset.mem_image, exists_prop, mem_boxes, mem_filter],
rintro _ ⟨J₁, h₁, rfl⟩ hne,
... | lemma | box_integral.prepartition.eventually_split_many_inf_eq_filter | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"exists_prop",
"filter",
"finset",
"finset.mem_image",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_split_many_inf_eq_filter_of_finite (s : set (prepartition I)) (hs : s.finite) :
∃ t : finset (ι × ℝ), ∀ π ∈ s,
π ⊓ (split_many I t) = (split_many I t).filter (λ J, ↑J ⊆ π.Union) | begin
have := λ π (hπ : π ∈ s), eventually_split_many_inf_eq_filter π,
exact (hs.eventually_all.2 this).exists
end | lemma | box_integral.prepartition.exists_split_many_inf_eq_filter_of_finite | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"filter",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_partition.exists_split_many_le {I : box ι} {π : prepartition I}
(h : is_partition π) : ∃ s, split_many I s ≤ π | (eventually_split_many_inf_eq_filter π).exists.imp $ λ s hs,
by { rwa [h.Union_eq, filter_of_true, inf_eq_right] at hs, exact λ J hJ, le_of_mem _ hJ } | lemma | box_integral.prepartition.is_partition.exists_split_many_le | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"inf_eq_right"
] | If `π` is a partition of `I`, then there exists a finite set `s` of hyperplanes such that
`split_many I s ≤ π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Union_eq_diff (π : prepartition I) :
∃ π' : prepartition I, π'.Union = I \ π.Union | begin
rcases π.eventually_split_many_inf_eq_filter.exists with ⟨s, hs⟩,
use (split_many I s).filter (λ J, ¬(J : set (ι → ℝ)) ⊆ π.Union),
simp [← hs]
end | lemma | box_integral.prepartition.exists_Union_eq_diff | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"filter"
] | For every prepartition `π` of `I` there exists a prepartition that covers exactly
`I \ π.Union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compl (π : prepartition I) : prepartition I | π.exists_Union_eq_diff.some | def | box_integral.prepartition.compl | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | If `π` is a prepartition of `I`, then `π.compl` is a prepartition of `I`
such that `π.compl.Union = I \ π.Union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Union_compl (π : prepartition I) : π.compl.Union = I \ π.Union | π.exists_Union_eq_diff.some_spec | lemma | box_integral.prepartition.Union_compl | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_congr {π₁ π₂ : prepartition I} (h : π₁.Union = π₂.Union) :
π₁.compl = π₂.compl | by { dunfold compl, congr' 1, rw h } | lemma | box_integral.prepartition.compl_congr | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | Since the definition of `box_integral.prepartition.compl` uses `Exists.some`,
the result depends only on `π.Union`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_partition.compl_eq_bot {π : prepartition I} (h : is_partition π) : π.compl = ⊥ | by rw [← Union_eq_empty, Union_compl, h.Union_eq, diff_self] | lemma | box_integral.prepartition.is_partition.compl_eq_bot | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_top : (⊤ : prepartition I).compl = ⊥ | (is_partition_top I).compl_eq_bot | lemma | box_integral.prepartition.compl_top | analysis.box_integral.partition | src/analysis/box_integral/partition/split.lean | [
"analysis.box_integral.partition.basic"
] | [
"compl_eq_bot",
"compl_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_center (I : box ι) : prepartition I | { boxes := finset.univ.map (box.split_center_box_emb I),
le_of_mem' := by simp [I.split_center_box_le],
pairwise_disjoint :=
begin
rw [finset.coe_map, finset.coe_univ, image_univ],
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ Hne,
exact I.disjoint_split_center_box (mt (congr_arg _) Hne)
end } | def | box_integral.prepartition.split_center | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [
"finset.coe_map",
"finset.coe_univ"
] | Split a box in `ℝⁿ` into `2 ^ n` boxes by hyperplanes passing through its center. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_split_center : J ∈ split_center I ↔ ∃ s, I.split_center_box s = J | by simp [split_center] | lemma | box_integral.prepartition.mem_split_center | 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 | |
is_partition_split_center (I : box ι) : is_partition (split_center I) | λ x hx, by simp [hx] | lemma | box_integral.prepartition.is_partition_split_center | 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 | |
upper_sub_lower_of_mem_split_center (h : J ∈ split_center I) (i : ι) :
J.upper i - J.lower i = (I.upper i - I.lower i) / 2 | let ⟨s, hs⟩ := mem_split_center.1 h in hs ▸ I.upper_sub_lower_split_center_box s i | lemma | box_integral.prepartition.upper_sub_lower_of_mem_split_center | 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 | |
subbox_induction_on {p : box ι → Prop} (I : box ι)
(H_ind : ∀ J ≤ I, (∀ J' ∈ split_center J, p J') → p J)
(H_nhds : ∀ z ∈ I.Icc, ∃ (U ∈ 𝓝[I.Icc] z), ∀ (J ≤ I) (m : ℕ), z ∈ J.Icc → J.Icc ⊆ U →
(∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
p I | begin
refine subbox_induction_on' I (λ J hle hs, H_ind J hle $ λ J' h', _) H_nhds,
rcases mem_split_center.1 h' with ⟨s, rfl⟩,
exact hs s
end | lemma | box_integral.box.subbox_induction_on | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [] | Let `p` be a predicate on `box ι`, let `I` be a box. Suppose that the following two properties
hold true.
* Consider a smaller box `J ≤ I`. The hyperplanes passing through the center of `J` split it into
`2 ^ n` boxes. If `p` holds true on each of these boxes, then it true on `J`.
* For each `z` in the closed box `I... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_tagged_partition_is_Henstock_is_subordinate_homothetic (I : box ι)
(r : (ι → ℝ) → Ioi (0 : ℝ)) :
∃ π : tagged_prepartition I, π.is_partition ∧ π.is_Henstock ∧ π.is_subordinate r ∧
(∀ J ∈ π, ∃ m : ℕ, ∀ i, (J : _).upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) ∧
π.distortion = I.distortion | begin
refine subbox_induction_on I (λ J hle hJ, _) (λ z hz, _),
{ choose! πi hP hHen hr Hn Hd using hJ, choose! n hn using Hn,
have hP : ((split_center J).bUnion_tagged πi).is_partition,
from (is_partition_split_center _).bUnion_tagged hP,
have hsub : ∀ (J' ∈ (split_center J).bUnion_tagged πi), ∃ n : ... | lemma | box_integral.box.exists_tagged_partition_is_Henstock_is_subordinate_homothetic | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [
"div_div",
"forall_eq",
"inter_mem_nhds_within",
"le_rfl",
"pow_succ",
"set.subset_inter_iff"
] | Given a box `I` in `ℝⁿ` and a function `r : ℝⁿ → (0, ∞)`, there exists a tagged partition `π` of
`I` such that
* `π` is a Henstock partition;
* `π` is subordinate to `r`;
* each box in `π` is homothetic to `I` with coefficient of the form `1 / 2 ^ m`.
This lemma implies that the Henstock filter is nontrivial, hence t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_tagged_le_is_Henstock_is_subordinate_Union_eq {I : box ι} (r : (ι → ℝ) → Ioi (0 : ℝ))
(π : prepartition I) :
∃ π' : tagged_prepartition I, π'.to_prepartition ≤ π ∧
π'.is_Henstock ∧ π'.is_subordinate r ∧ π'.distortion = π.distortion ∧
π'.Union = π.Union | begin
have := λ J, box.exists_tagged_partition_is_Henstock_is_subordinate_homothetic J r,
choose! πi πip πiH πir hsub πid, clear hsub,
refine ⟨π.bUnion_tagged πi, bUnion_le _ _, is_Henstock_bUnion_tagged.2 (λ J _, πiH J),
is_subordinate_bUnion_tagged.2 (λ J _, πir J), _, π.Union_bUnion_partition (λ J _, πip J... | lemma | box_integral.prepartition.exists_tagged_le_is_Henstock_is_subordinate_Union_eq | analysis.box_integral.partition | src/analysis/box_integral/partition/subbox_induction.lean | [
"analysis.box_integral.box.subbox_induction",
"analysis.box_integral.partition.tagged"
] | [] | Given a box `I` in `ℝⁿ`, a function `r : ℝⁿ → (0, ∞)`, and a prepartition `π` of `I`, there
exists a tagged prepartition `π'` of `I` such that
* each box of `π'` is included in some box of `π`;
* `π'` is a Henstock partition;
* `π'` is subordinate to `r`;
* `π'` covers exactly the same part of `I` as `π`;
* the distor... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_subordinate (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) : tagged_prepartition I | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some | def | box_integral.prepartition.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"
] | [] | Given a prepartition `π` of a box `I` and a function `r : ℝⁿ → (0, ∞)`, `π.to_subordinate r`
is a tagged partition `π'` such that
* each box of `π'` is included in some box of `π`;
* `π'` is a Henstock partition;
* `π'` is subordinate to `r`;
* `π'` covers exactly the same part of `I` as `π`;
* the distortion of `π'` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_subordinate_to_prepartition_le (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π.to_subordinate r).to_prepartition ≤ π | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some_spec.1 | lemma | box_integral.prepartition.to_subordinate_to_prepartition_le | 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 | |
is_Henstock_to_subordinate (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π.to_subordinate r).is_Henstock | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some_spec.2.1 | lemma | box_integral.prepartition.is_Henstock_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 | |
is_subordinate_to_subordinate (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π.to_subordinate r).is_subordinate r | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some_spec.2.2.1 | lemma | box_integral.prepartition.is_subordinate_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 | |
distortion_to_subordinate (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π.to_subordinate r).distortion = π.distortion | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some_spec.2.2.2.1 | lemma | box_integral.prepartition.distortion_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 | |
Union_to_subordinate (π : prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π.to_subordinate r).Union = π.Union | (π.exists_tagged_le_is_Henstock_is_subordinate_Union_eq r).some_spec.2.2.2.2 | lemma | box_integral.prepartition.Union_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 | |
union_compl_to_subordinate (π₁ : tagged_prepartition I) (π₂ : prepartition I)
(hU : π₂.Union = I \ π₁.Union) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
tagged_prepartition I | π₁.disj_union (π₂.to_subordinate r)
(((π₂.Union_to_subordinate r).trans hU).symm ▸ disjoint_sdiff_self_right) | def | box_integral.tagged_prepartition.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"
] | [
"disjoint_sdiff_self_right"
] | Given a tagged prepartition `π₁`, a prepartition `π₂` that covers exactly `I \ π₁.Union`, and
a function `r : ℝⁿ → (0, ∞)`, returns the union of `π₁` and `π₂.to_subordinate r`. This partition
`π` has the following properties:
* `π` is a partition, i.e. it covers the whole `I`;
* `π₁.boxes ⊆ π.boxes`;
* `π.tag J = π₁.t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_partition_union_compl_to_subordinate (π₁ : tagged_prepartition I) (π₂ : prepartition I)
(hU : π₂.Union = I \ π₁.Union) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
is_partition (π₁.union_compl_to_subordinate π₂ hU r) | prepartition.is_partition_disj_union_of_eq_diff ((π₂.Union_to_subordinate r).trans hU) | lemma | box_integral.tagged_prepartition.is_partition_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 | |
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).boxes = π₁.boxes ∪ (π₂.to_subordinate r).boxes | rfl | lemma | box_integral.tagged_prepartition.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 |
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