Documentation

LeanPool.AnomalousDiffusion.Homogenization.Geometry.BoundaryLayer

Boundary Layer #

Support for the Armstrong–Vicol anomalous-diffusion formalization.

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def Homogenization.cubeBoundary {d : ℕ} (Q : TriadicCube d) :
Set (Vec d)

The geometric boundary of a half-open cube, encoded as the difference between the half-open realization and its open core.

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    def Homogenization.cubeShrunkSet {d : ℕ} (Q : TriadicCube d) (t : ℝ) :
    Set (Vec d)

    The cube obtained by shrinking each face inward by the normalized amount t. For t = 0 this recovers the original half-open cube.

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      def Homogenization.cubeBoundaryLayer {d : ℕ} (Q : TriadicCube d) (t : ℝ) :
      Set (Vec d)

      The boundary layer of thickness t, defined as the part of the cube left after removing the shrunk core.

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        @[simp]
        theorem Homogenization.mem_cubeBoundary_iff {d : ℕ} {Q : TriadicCube d} {x : Vec d} :
        x ∈ cubeBoundary Q ↔ x ∈ cubeSet Q ∧ x ∉ openCubeSet Q
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        @[simp]
        theorem Homogenization.mem_cubeShrunkSet_iff {d : ℕ} {Q : TriadicCube d} {t : ℝ} {x : Vec d} :
        x ∈ cubeShrunkSet Q t ↔ ∀ (i : Fin d), (↑(Q.index i) - 1 / 2 + t) * cubeScaleFactor Q ≤ x i ∧ x i < (↑(Q.index i) + 1 / 2 - t) * cubeScaleFactor Q
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        @[simp]
        theorem Homogenization.mem_cubeBoundaryLayer_iff {d : ℕ} {Q : TriadicCube d} {t : ℝ} {x : Vec d} :
        x ∈ cubeBoundaryLayer Q t ↔ x ∈ cubeSet Q ∧ x ∉ cubeShrunkSet Q t
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        @[simp]
        theorem Homogenization.cubeShrunkSet_zero {d : ℕ} (Q : TriadicCube d) :
        cubeShrunkSet Q 0 = cubeSet Q
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        @[simp]
        theorem Homogenization.cubeBoundaryLayer_zero {d : ℕ} (Q : TriadicCube d) :
        cubeBoundaryLayer Q 0 = ∅
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        theorem Homogenization.BoundaryLayer.cubeScaleFactor_pos {d : ℕ} (Q : TriadicCube d) :
        0 < cubeScaleFactor Q
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        theorem Homogenization.cubeBoundary_subset_cubeSet {d : ℕ} (Q : TriadicCube d) :
        cubeBoundary Q ⊆ cubeSet Q
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        theorem Homogenization.cubeBoundaryLayer_subset_cubeSet {d : ℕ} (Q : TriadicCube d) (t : ℝ) :
        cubeBoundaryLayer Q t ⊆ cubeSet Q
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        theorem Homogenization.cubeShrunkSet_anti {d : ℕ} (Q : TriadicCube d) :
        Antitone (cubeShrunkSet Q)
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        theorem Homogenization.cubeShrunkSet_subset_cubeSet {d : ℕ} (Q : TriadicCube d) {t : ℝ} (ht : 0 ≤ t) :
        cubeShrunkSet Q t ⊆ cubeSet Q
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        theorem Homogenization.cubeBoundaryLayer_mono {d : ℕ} (Q : TriadicCube d) :
        Monotone (cubeBoundaryLayer Q)
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        theorem Homogenization.openCubeSet_subset_cubeSet {d : ℕ} (Q : TriadicCube d) :
        openCubeSet Q ⊆ cubeSet Q
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        theorem Homogenization.pairwiseDisjoint_openCubeSet_descendantsAtDepth {d : ℕ} (Q : TriadicCube d) (n : ℕ) :
        (↑(descendantsAtDepth Q n)).PairwiseDisjoint openCubeSet
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        theorem Homogenization.openCubeSet_union_cubeBoundary_eq_cubeSet {d : ℕ} (Q : TriadicCube d) :
        openCubeSet Q ∪ cubeBoundary Q = cubeSet Q
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        theorem Homogenization.cubeShrunkSet_eq_empty_of_half_le {d : ℕ} [NeZero d] (Q : TriadicCube d) {t : ℝ} (ht : 1 / 2 ≤ t) :
        cubeShrunkSet Q t = ∅
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        theorem Homogenization.cubeBoundaryLayer_eq_cubeSet_of_half_le {d : ℕ} [NeZero d] (Q : TriadicCube d) {t : ℝ} (ht : 1 / 2 ≤ t) :
        cubeBoundaryLayer Q t = cubeSet Q
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        theorem Homogenization.center_mem_cubeShrunkSet_of_lt_half {d : ℕ} (Q : TriadicCube d) {t : ℝ} (ht : t < 1 / 2) :
        (fun (i : Fin d) => ↑(Q.index i) * cubeScaleFactor Q) ∈ cubeShrunkSet Q t
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        theorem Homogenization.cubeShrunkSet_nonempty_of_lt_half {d : ℕ} (Q : TriadicCube d) {t : ℝ} (ht : t < 1 / 2) :
        (cubeShrunkSet Q t).Nonempty
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        theorem Homogenization.cubeShrunkSet_nonempty_iff {d : ℕ} [NeZero d] {Q : TriadicCube d} {t : ℝ} :
        (cubeShrunkSet Q t).Nonempty ↔ t < 1 / 2
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        theorem Homogenization.cubeShrunkSet_eq_empty_iff {d : ℕ} [NeZero d] {Q : TriadicCube d} {t : ℝ} :
        cubeShrunkSet Q t = ∅ ↔ 1 / 2 ≤ t
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        theorem Homogenization.cubeSet_middleChild_subset_cubeShrunkSet {d : ℕ} (Q : TriadicCube d) {t : ℝ} (ht : t ≤ 1 / 3) :
        cubeSet { scale := Q.scale - 1, index := fun (i : Fin d) => 3 * Q.index i } ⊆ cubeShrunkSet Q t
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        theorem Homogenization.cubeSet_childCube_subset_cubeShrunkSet_of_digits_eq_one {d : ℕ} (Q : TriadicCube d) (digits : Fin d → Fin 3) (hdigits : ∀ (i : Fin d), digits i = 1) {t : ℝ} (ht : t ≤ 1 / 3) :
        cubeSet { scale := Q.scale - 1, index := fun (i : Fin d) => 3 * Q.index i + ↑↑(digits i) - 1 } ⊆ cubeShrunkSet Q t
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        @[simp]
        theorem Homogenization.descendantsAtScale_pred {d : ℕ} (Q : TriadicCube d) :
        descendantsAtScale Q (Q.scale - 1) = childCubes Q
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        theorem Homogenization.middleChild_mem_childCubes {d : ℕ} (Q : TriadicCube d) :
        { scale := Q.scale - 1, index := fun (i : Fin d) => 3 * Q.index i } ∈ childCubes Q
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        theorem Homogenization.middleChild_mem_descendantsAtScale_pred {d : ℕ} (Q : TriadicCube d) :
        { scale := Q.scale - 1, index := fun (i : Fin d) => 3 * Q.index i } ∈ descendantsAtScale Q (Q.scale - 1)
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        theorem Homogenization.descendantsAtScale_pred_card {d : ℕ} (Q : TriadicCube d) :
        (descendantsAtScale Q (Q.scale - 1)).card = 3 ^ d
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        theorem Homogenization.cubeBoundaryLayer_subset_iUnion_childCubes_except_middle {d : ℕ} (Q : TriadicCube d) {t : ℝ} (ht : t ≤ 1 / 3) :
        cubeBoundaryLayer Q t ⊆ ⋃ R ∈ {S : TriadicCube d | S ∈ childCubes Q ∧ S ≠ { scale := Q.scale - 1, index := fun (i : Fin d) => 3 * Q.index i }}, cubeSet R