Documentation

Mathlib.Analysis.BoxIntegral.Partition.Filter

Filters used in box-based integrals #

First we define a structure BoxIntegral.IntegrationParams. This structure will be used as an argument in the definition of BoxIntegral.integral in order to use the same definition for a few well-known definitions of integrals based on partitions of a rectangular box into subboxes (Riemann integral, Henstock-Kurzweil integral, and McShane integral).

This structure holds three boolean values (see below), and encodes eight different sets of parameters; only four of these values are used somewhere in mathlib4. Three of them correspond to the integration theories listed above, and one is a generalization of the one-dimensional Henstock-Kurzweil integral such that the divergence theorem works without additional integrability assumptions.

Finally, for each set of parameters l : BoxIntegral.IntegrationParams and a rectangular box I : BoxIntegral.Box ι, we define several Filters that will be used either in the definition of the corresponding integral, or in the proofs of its properties. We equip BoxIntegral.IntegrationParams with a BoundedOrder structure such that larger IntegrationParams produce larger filters.

Main definitions #

Integration parameters #

The structure BoxIntegral.IntegrationParams has 3 boolean fields with the following meaning:

Well-known sets of parameters #

Out of eight possible values of BoxIntegral.IntegrationParams, the following four are used in the library.

Filters and predicates on TaggedPrepartition I #

For each value of IntegrationParams and a rectangular box I, we define a few filters on TaggedPrepartition I. First, we define a predicate

structure BoxIntegral.IntegrationParams.MemBaseSet (l : BoxIntegral.IntegrationParams)
  (I : BoxIntegral.Box ι) (c : ℝ≥0) (r : (ι → ℝ) → Ioi (0 : ℝ))
  (π : BoxIntegral.TaggedPrepartition I) : Prop where

This predicate says that

The last condition is always true for c > 1, see TODO section for more details.

Then we define a predicate BoxIntegral.IntegrationParams.RCond on functions r : (ι → ℝ) → {x : ℝ | 0 < x}. If l.bRiemann, then this predicate requires r to be a constant function, otherwise it imposes no restrictions on r. We introduce this definition to prove a few dot-notation lemmas: e.g., BoxIntegral.IntegrationParams.RCond.min says that the pointwise minimum of two functions that satisfy this condition satisfies this condition as well.

Then we define four filters on BoxIntegral.TaggedPrepartition I.

Implementation details #

TODO #

Currently, BoxIntegral.IntegrationParams.MemBaseSet explicitly requires that there exists a partition of the complement I \ π.iUnion with distortion ≤ c. For c > 1, this condition is always true but the proof of this fact requires more API about BoxIntegral.Prepartition.splitMany. We should formalize this fact, then either require c > 1 everywhere, or replace ≤ c with < c so that we automatically get c > 1 for a non-trivial prepartition (and consider the special case π = ⊥ separately if needed).

Tags #

integral, rectangular box, partition, filter

theorem BoxIntegral.IntegrationParams.ext_iff {x : BoxIntegral.IntegrationParams} {y : BoxIntegral.IntegrationParams} :
x = y x.bRiemann = y.bRiemann x.bHenstock = y.bHenstock x.bDistortion = y.bDistortion
theorem BoxIntegral.IntegrationParams.ext {x : BoxIntegral.IntegrationParams} {y : BoxIntegral.IntegrationParams} (bRiemann : x.bRiemann = y.bRiemann) (bHenstock : x.bHenstock = y.bHenstock) (bDistortion : x.bDistortion = y.bDistortion) :
x = y

An IntegrationParams 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 true means that the filter corresponds to a Riemann-style integral, i.e. in the definition of integrability we require a constant upper estimate r on the size of boxes of a tagged partition; the value false means that the estimate may depend on the position of the tag.

  • bHenstock: the value true means that we require that each tag belongs to its own closed box; the value false means that we only require that tags belong to the ambient box.

  • bDistortion: the value true means that r can depend on the maximal ratio of sides of the same box of a partition. Presence of this case makes quite a few proofs harder but we can prove the divergence theorem only for the filter BoxIntegral.IntegrationParams.GP = ⊥ = {bRiemann := false, bHenstock := true, bDistortion := true}.

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    Auxiliary equivalence with a product type used to lift an order.

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    • One or more equations did not get rendered due to their size.
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      The value BoxIntegral.IntegrationParams.GP = ⊥ (bRiemann = false, bHenstock = true, bDistortion = true) corresponds to a generalization of the Henstock integral such that the Divergence theorem holds true without additional integrability assumptions, see the module docstring for details.

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      • One or more equations did not get rendered due to their size.
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      • x✝.instDecidableEq x = decidable_of_iff (x✝.bRiemann = x.bRiemann x✝.bHenstock = x.bHenstock x✝.bDistortion = x.bDistortion)

      The BoxIntegral.IntegrationParams 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 box.

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        The BoxIntegral.IntegrationParams 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.

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          The BoxIntegral.IntegrationParams 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).

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            The BoxIntegral.IntegrationParams 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 distortion of all boxes of the partition.

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              The predicate corresponding to a base set of the filter defined by an IntegrationParams. 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 than or equal to c;
              • if l.bDistortion, then there exists a prepartition π' with distortion ≤ c that covers exactly I \ π.iUnion.

              The last condition is automatically verified for partitions, and is used in the proof of the Sacks-Henstock inequality to compare two prepartitions covering the same part of the box.

              It is also automatically satisfied for any c > 1, see TODO section of the module docstring for details.

              • isSubordinate : π.IsSubordinate r
              • isHenstock : l.bHenstock = trueπ.IsHenstock
              • distortion_le : l.bDistortion = trueπ.distortion c
              • exists_compl : l.bDistortion = true∃ (π' : BoxIntegral.Prepartition I), π'.iUnion = I \ π.iUnion π'.distortion c
              Instances For
                theorem BoxIntegral.IntegrationParams.MemBaseSet.isSubordinate {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : NNReal} {r : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π) :
                π.IsSubordinate r
                theorem BoxIntegral.IntegrationParams.MemBaseSet.isHenstock {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : NNReal} {r : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π) :
                l.bHenstock = trueπ.IsHenstock
                theorem BoxIntegral.IntegrationParams.MemBaseSet.distortion_le {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : NNReal} {r : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π) :
                l.bDistortion = trueπ.distortion c
                theorem BoxIntegral.IntegrationParams.MemBaseSet.exists_compl {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : NNReal} {r : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π) :
                l.bDistortion = true∃ (π' : BoxIntegral.Prepartition I), π'.iUnion = I \ π.iUnion π'.distortion c

                A predicate saying that in case l.bRiemann = true, the function r is a constant.

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                • l.RCond r = (l.bRiemann = true∀ (x : ι), r x = r 0)
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                  A set s : Set (TaggedPrepartition I) belongs to l.toFilterDistortion I c if there exists a function r : ℝⁿ → (0, ∞) (or a constant r if l.bRiemann = true) such that s contains each prepartition π such that l.MemBaseSet I c r π.

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                    A set s : Set (TaggedPrepartition I) belongs to l.toFilter I if for any c : ℝ≥0 there exists a function r : ℝⁿ → (0, ∞) (or a constant r if l.bRiemann = true) such that s contains each prepartition π such that l.MemBaseSet I c r π.

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                    • l.toFilter I = ⨆ (c : NNReal), l.toFilterDistortion I c
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                      A set s : Set (TaggedPrepartition I) belongs to l.toFilterDistortioniUnion I c π₀ if there exists a function r : ℝⁿ → (0, ∞) (or a constant r if l.bRiemann = true) such that s contains each prepartition π such that l.MemBaseSet I c r π and π.iUnion = π₀.iUnion.

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                        A set s : Set (TaggedPrepartition I) belongs to l.toFilteriUnion I π₀ if for any c : ℝ≥0 there exists a function r : ℝⁿ → (0, ∞) (or a constant r if l.bRiemann = true) such that s contains each prepartition π such that l.MemBaseSet I c r π and π.iUnion = π₀.iUnion.

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                          theorem BoxIntegral.IntegrationParams.rCond_of_bRiemann_eq_false {ι : Type u_2} (l : BoxIntegral.IntegrationParams) (hl : l.bRiemann = false) {r : (ι)(Set.Ioi 0)} :
                          l.RCond r
                          theorem BoxIntegral.IntegrationParams.MemBaseSet.mono' {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) {c₁ : NNReal} {c₂ : NNReal} {l₁ : BoxIntegral.IntegrationParams} {l₂ : BoxIntegral.IntegrationParams} {r₁ : (ι)(Set.Ioi 0)} {r₂ : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (h : l₁ l₂) (hc : c₁ c₂) (hr : Jπ, r₁ (π.tag J) r₂ (π.tag J)) (hπ : l₁.MemBaseSet I c₁ r₁ π) :
                          l₂.MemBaseSet I c₂ r₂ π
                          theorem BoxIntegral.IntegrationParams.MemBaseSet.mono {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) {c₁ : NNReal} {c₂ : NNReal} {l₁ : BoxIntegral.IntegrationParams} {l₂ : BoxIntegral.IntegrationParams} {r₁ : (ι)(Set.Ioi 0)} {r₂ : (ι)(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (h : l₁ l₂) (hc : c₁ c₂) (hr : xBoxIntegral.Box.Icc I, r₁ x r₂ x) (hπ : l₁.MemBaseSet I c₁ r₁ π) :
                          l₂.MemBaseSet I c₂ r₂ π
                          theorem BoxIntegral.IntegrationParams.MemBaseSet.exists_common_compl {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c₁ : NNReal} {c₂ : NNReal} {l : BoxIntegral.IntegrationParams} {r₁ : (ι)(Set.Ioi 0)} {r₂ : (ι)(Set.Ioi 0)} {π₁ : BoxIntegral.TaggedPrepartition I} {π₂ : BoxIntegral.TaggedPrepartition I} (h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂) (hU : π₁.iUnion = π₂.iUnion) :
                          ∃ (π : BoxIntegral.Prepartition I), π.iUnion = I \ π₁.iUnion (l.bDistortion = trueπ.distortion c₁) (l.bDistortion = trueπ.distortion c₂)
                          theorem BoxIntegral.IntegrationParams.MemBaseSet.unionComplToSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : NNReal} {l : BoxIntegral.IntegrationParams} {r₁ : (ι)(Set.Ioi 0)} {r₂ : (ι)(Set.Ioi 0)} {π₁ : BoxIntegral.TaggedPrepartition I} (hπ₁ : l.MemBaseSet I c r₁ π₁) (hle : xBoxIntegral.Box.Icc I, r₂ x r₁ x) {π₂ : BoxIntegral.Prepartition I} (hU : π₂.iUnion = I \ π₁.iUnion) (hc : l.bDistortion = trueπ₂.distortion c) :
                          l.MemBaseSet I c r₁ (π₁.unionComplToSubordinate π₂ hU r₂)
                          theorem BoxIntegral.IntegrationParams.MemBaseSet.filter {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : NNReal} {l : BoxIntegral.IntegrationParams} {π : BoxIntegral.TaggedPrepartition I} {r : (ι)(Set.Ioi 0)} (hπ : l.MemBaseSet I c r π) (p : BoxIntegral.Box ιProp) :
                          l.MemBaseSet I c r (π.filter p)
                          theorem BoxIntegral.IntegrationParams.biUnionTagged_memBaseSet {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : NNReal} {l : BoxIntegral.IntegrationParams} {r : (ι)(Set.Ioi 0)} {π : BoxIntegral.Prepartition I} {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J} (h : Jπ, l.MemBaseSet J c r (πi J)) (hp : Jπ, (πi J).IsPartition) (hc : l.bDistortion = trueπ.compl.distortion c) :
                          l.MemBaseSet I c r (π.biUnionTagged πi)
                          theorem BoxIntegral.IntegrationParams.RCond.mono {l₁ : BoxIntegral.IntegrationParams} {l₂ : BoxIntegral.IntegrationParams} {ι : Type u_2} {r : (ι)(Set.Ioi 0)} (h : l₁ l₂) (hr : l₂.RCond r) :
                          l₁.RCond r
                          theorem BoxIntegral.IntegrationParams.RCond.min {l : BoxIntegral.IntegrationParams} {ι : Type u_2} {r₁ : (ι)(Set.Ioi 0)} {r₂ : (ι)(Set.Ioi 0)} (h₁ : l.RCond r₁) (h₂ : l.RCond r₂) :
                          l.RCond fun (x : ι) => min (r₁ x) (r₂ x)
                          theorem BoxIntegral.IntegrationParams.toFilterDistortion_mono {ι : Type u_1} [Fintype ι] {c₁ : NNReal} {c₂ : NNReal} {l₁ : BoxIntegral.IntegrationParams} {l₂ : BoxIntegral.IntegrationParams} (I : BoxIntegral.Box ι) (h : l₁ l₂) (hc : c₁ c₂) :
                          l₁.toFilterDistortion I c₁ l₂.toFilterDistortion I c₂
                          theorem BoxIntegral.IntegrationParams.toFilter_mono {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) {l₁ : BoxIntegral.IntegrationParams} {l₂ : BoxIntegral.IntegrationParams} (h : l₁ l₂) :
                          l₁.toFilter I l₂.toFilter I
                          theorem BoxIntegral.IntegrationParams.hasBasis_toFilterDistortion {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (c : NNReal) :
                          (l.toFilterDistortion I c).HasBasis l.RCond fun (r : (ι)(Set.Ioi 0)) => {π : BoxIntegral.TaggedPrepartition I | l.MemBaseSet I c r π}
                          theorem BoxIntegral.IntegrationParams.hasBasis_toFilterDistortioniUnion {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (c : NNReal) (π₀ : BoxIntegral.Prepartition I) :
                          (l.toFilterDistortioniUnion I c π₀).HasBasis l.RCond fun (r : (ι)(Set.Ioi 0)) => {π : BoxIntegral.TaggedPrepartition I | l.MemBaseSet I c r π π.iUnion = π₀.iUnion}
                          theorem BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I) :
                          (BoxIntegral.IntegrationParams.toFilteriUnion I π₀).HasBasis (fun (r : NNReal(ι)(Set.Ioi 0)) => ∀ (c : NNReal), l.RCond (r c)) fun (r : NNReal(ι)(Set.Ioi 0)) => {π : BoxIntegral.TaggedPrepartition I | ∃ (c : NNReal), l.MemBaseSet I c (r c) π π.iUnion = π₀.iUnion}
                          theorem BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion_top {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) :
                          (BoxIntegral.IntegrationParams.toFilteriUnion I ).HasBasis (fun (r : NNReal(ι)(Set.Ioi 0)) => ∀ (c : NNReal), l.RCond (r c)) fun (r : NNReal(ι)(Set.Ioi 0)) => {π : BoxIntegral.TaggedPrepartition I | ∃ (c : NNReal), l.MemBaseSet I c (r c) π π.IsPartition}
                          theorem BoxIntegral.IntegrationParams.hasBasis_toFilter {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) :
                          (l.toFilter I).HasBasis (fun (r : NNReal(ι)(Set.Ioi 0)) => ∀ (c : NNReal), l.RCond (r c)) fun (r : NNReal(ι)(Set.Ioi 0)) => {π : BoxIntegral.TaggedPrepartition I | ∃ (c : NNReal), l.MemBaseSet I c (r c) π}
                          theorem BoxIntegral.IntegrationParams.exists_memBaseSet_le_iUnion_eq {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : NNReal} (l : BoxIntegral.IntegrationParams) (π₀ : BoxIntegral.Prepartition I) (hc₁ : π₀.distortion c) (hc₂ : π₀.compl.distortion c) (r : (ι)(Set.Ioi 0)) :
                          ∃ (π : BoxIntegral.TaggedPrepartition I), l.MemBaseSet I c r π π.toPrepartition π₀ π.iUnion = π₀.iUnion
                          theorem BoxIntegral.IntegrationParams.exists_memBaseSet_isPartition {ι : Type u_1} [Fintype ι] {c : NNReal} (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (hc : I.distortion c) (r : (ι)(Set.Ioi 0)) :
                          ∃ (π : BoxIntegral.TaggedPrepartition I), l.MemBaseSet I c r π π.IsPartition
                          theorem BoxIntegral.IntegrationParams.toFilterDistortioniUnion_neBot {ι : Type u_1} [Fintype ι] {c : NNReal} (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I) (hc₁ : π₀.distortion c) (hc₂ : π₀.compl.distortion c) :
                          (l.toFilterDistortioniUnion I c π₀).NeBot
                          instance BoxIntegral.IntegrationParams.toFilterDistortioniUnion_neBot' {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I) :
                          (l.toFilterDistortioniUnion I (max π₀.distortion π₀.compl.distortion) π₀).NeBot
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                          instance BoxIntegral.IntegrationParams.toFilterDistortion_neBot {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) :
                          (l.toFilterDistortion I I.distortion).NeBot
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                          instance BoxIntegral.IntegrationParams.toFilter_neBot {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) :
                          (l.toFilter I).NeBot
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