| import OAI.Analysis.Brenier.Argmax |
| import OAI.Analysis.Brenier.Uniform |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
|
|
| namespace Problem358 |
| namespace CellGeometry |
|
|
| variable {d : ℕ} {ι : Type*} |
|
|
| /-- The closed cell of one affine branch, including ties. -/ |
| def closedCell (K : Set (E d)) (y : ι → E d) (c : ι → ℝ) (i : ι) : Set (E d) := |
| {x | x ∈ K ∧ ∀ j, inner ℝ x (y j) + c j ≤ inner ℝ x (y i) + c i} |
|
|
| theorem closedCell_subset (K : Set (E d)) (y : ι → E d) (c : ι → ℝ) (i : ι) : |
| closedCell K y c i ⊆ K := fun _ h => h.1 |
|
|
| theorem isClosed_closedCell {K : Set (E d)} (hK : IsClosed K) |
| (y : ι → E d) (c : ι → ℝ) (i : ι) : IsClosed (closedCell K y c i) := by |
| have heq : closedCell K y c i = K ∩ ⋂ j, |
| {x | inner ℝ x (y j) + c j ≤ inner ℝ x (y i) + c i} := by |
| ext x |
| simp [closedCell] |
| rw [heq] |
| exact hK.inter (isClosed_iInter fun j => |
| isClosed_le ((continuous_id.inner continuous_const).add continuous_const) |
| ((continuous_id.inner continuous_const).add continuous_const)) |
|
|
| theorem isCompact_closedCell {K : Set (E d)} (hK : IsCompact K) |
| (y : ι → E d) (c : ι → ℝ) (i : ι) : IsCompact (closedCell K y c i) := |
| hK.of_isClosed_subset (isClosed_closedCell hK.isClosed y c i) |
| (closedCell_subset K y c i) |
|
|
| theorem convex_closedCell {K : Set (E d)} (hK : Convex ℝ K) |
| (y : ι → E d) (c : ι → ℝ) (i : ι) : Convex ℝ (closedCell K y c i) := by |
| intro x hx z hz a b ha hb hab |
| refine ⟨hK hx.1 hz.1 ha hb hab, ?_⟩ |
| intro j |
| simp only [inner_add_left, inner_smul_left, starRingEnd_apply, star_trivial] |
| have h1 := mul_le_mul_of_nonneg_left (hx.2 j) ha |
| have h2 := mul_le_mul_of_nonneg_left (hz.2 j) hb |
| nlinarith [congrArg (fun t : ℝ => t * c i) hab, |
| congrArg (fun t : ℝ => t * c j) hab] |
|
|
| theorem exists_mem_closedCell [Fintype ι] [Nonempty ι] |
| {K : Set (E d)} (y : ι → E d) (c : ι → ℝ) {x : E d} (hx : x ∈ K) : |
| ∃ i, x ∈ closedCell K y c i := by |
| obtain ⟨i, _, hi⟩ := Finset.exists_max_image Finset.univ |
| (fun j => inner ℝ x (y j) + c j) Finset.univ_nonempty |
| exact ⟨i, hx, fun j => hi j (Finset.mem_univ j)⟩ |
|
|
| theorem iUnion_closedCell [Fintype ι] [Nonempty ι] |
| (K : Set (E d)) (y : ι → E d) (c : ι → ℝ) : |
| (⋃ i, closedCell K y c i) = K := by |
| ext x |
| constructor |
| · intro hx |
| obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx |
| exact hi.1 |
| · intro hx |
| obtain ⟨i, hi⟩ := exists_mem_closedCell y c hx |
| exact Set.mem_iUnion.mpr ⟨i, hi⟩ |
|
|
| /-- Positive-volume cells have strictly active points in the interior, once affine ties |
| have been removed as a null set. -/ |
| theorem exists_strict_active_interior {K : Set (E d)} |
| (hconv : Convex ℝ K) (y : ι → E d) (c : ι → ℝ) (i : ι) |
| (hpos : volume (closedCell K y c i) ≠ 0) |
| (hties : ∀ᵐ x ∂volume, ∀ j, j ≠ i → |
| inner ℝ x (y j) + c j ≠ inner ℝ x (y i) + c i) : |
| ∃ x ∈ interior K, ∀ j, j ≠ i → |
| inner ℝ x (y j) + c j < inner ℝ x (y i) + c i := by |
| have hboundary : ∀ᵐ x ∂(volume : Measure (E d)), x ∉ frontier K := by |
| simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using hconv.addHaar_frontier volume |
| have hgood : ∀ᵐ x ∂volume.restrict (closedCell K y c i), |
| x ∉ frontier K ∧ ∀ j, j ≠ i → |
| inner ℝ x (y j) + c j ≠ inner ℝ x (y i) + c i := |
| ae_restrict_of_ae (hboundary.and hties) |
| obtain ⟨x, hx, hxb, hxt⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos hgood |
| refine ⟨x, ?_, fun j hj => lt_of_le_of_ne (hx.2 j) (hxt j hj)⟩ |
| by_contra hn |
| exact hxb ⟨subset_closure hx.1, hn⟩ |
|
|
| /-- Convex interpolation of the intercepts and source points remains in the same cell. -/ |
| theorem combo_mem_closedCell {K : Set (E d)} (hK : Convex ℝ K) |
| (y : ι → E d) (c e : ι → ℝ) (i : ι) {x z : E d} |
| (hx : x ∈ closedCell K y c i) (hz : z ∈ closedCell K y e i) |
| {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) : |
| a • x + b • z ∈ closedCell K y (fun j => a * c j + b * e j) i := by |
| refine ⟨hK hx.1 hz.1 ha hb hab, ?_⟩ |
| intro j |
| simp only [inner_add_left, inner_smul_left, starRingEnd_apply, star_trivial] |
| have h1 := mul_le_mul_of_nonneg_left (hx.2 j) ha |
| have h2 := mul_le_mul_of_nonneg_left (hz.2 j) hb |
| nlinarith |
|
|
| /-- The opposite-intercept midpoint inclusion used in the moment estimate. -/ |
| theorem midpoint_mem_closedCell {K : Set (E d)} (hK : Convex ℝ K) |
| (y : ι → E d) (c q : ι → ℝ) (i : ι) (s : ℝ) {x z : E d} |
| (hx : x ∈ closedCell K y (fun j => c j - s * q j) i) |
| (hz : z ∈ closedCell K y (fun j => c j + s * q j) i) : |
| (1 / 2 : ℝ) • x + (1 / 2 : ℝ) • z ∈ closedCell K y c i := by |
| have h := combo_mem_closedCell hK y _ _ i hx hz |
| (a := (1 / 2 : ℝ)) (b := (1 / 2 : ℝ)) |
| (by norm_num) (by norm_num) (by norm_num) |
| have heq : (fun j => (1 / 2 : ℝ) * (c j - s * q j) + |
| (1 / 2 : ℝ) * (c j + s * q j)) = c := by |
| funext j |
| ring |
| rwa [heq] at h |
|
|
| /-- An affine equality plane separating two strictly active interior points meets |
| the interior of the convex source. -/ |
| theorem exists_affine_equality_interior {K : Set (E d)} (hK : Convex ℝ K) |
| {y z : E d} {c e : ℝ} {p q : E d} |
| (hp : p ∈ interior K) (hq : q ∈ interior K) |
| (hp_lt : inner ℝ p y + c < inner ℝ p z + e) |
| (hq_lt : inner ℝ q z + e < inner ℝ q y + c) : |
| ∃ x ∈ interior K, inner ℝ x y + c = inner ℝ x z + e := by |
| let f : E d → ℝ := fun x => (inner ℝ x y + c) - (inner ℝ x z + e) |
| have hf : Continuous f := |
| ((continuous_id.inner continuous_const).add continuous_const).sub |
| ((continuous_id.inner continuous_const).add continuous_const) |
| have hzero : (0 : ℝ) ∈ Set.Icc (f p) (f q) := by |
| constructor <;> dsimp [f] <;> linarith |
| obtain ⟨x, hx, heq⟩ := hK.interior.isPreconnected.intermediate_value |
| hp hq hf.continuousOn hzero |
| exact ⟨x, hx, sub_eq_zero.mp heq⟩ |
|
|
| end CellGeometry |
| end Problem358 |
|
|
| end |
|
|
| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators InnerProductSpace |
|
|
| namespace Problem358 |
| namespace CellGeometry |
|
|
| variable {d : ℕ} {ι : Type*} |
|
|
| /-- Distinct-site closed cells overlap only along null affine tie planes. -/ |
| theorem pairwise_aedisjoint_closedCell {ρ : Measure (E d)} (hρ : ρ ≪ volume) |
| (K : Set (E d)) (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) : |
| Pairwise (fun i j => AEDisjoint ρ (closedCell K y c i) (closedCell K y c j)) := by |
| intro i j hij |
| change ρ (closedCell K y c i ∩ closedCell K y c j) = 0 |
| apply hρ |
| apply measure_mono_null (t := {x | inner ℝ x (y i) + c i = inner ℝ x (y j) + c j}) |
| · intro x hx |
| exact le_antisymm (hx.2.2 i) (hx.1.2 j) |
| · exact haar_affine_tie_null volume (hy.ne hij) (c i) (c j) |
|
|
| /-- The closed cells form a finite partition for all measures absolutely continuous |
| with respect to volume. -/ |
| theorem sum_measure_closedCell [Fintype ι] [Nonempty ι] |
| {ρ : Measure (E d)} (hρ : ρ ≪ volume) {K : Set (E d)} (hK : IsClosed K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) : |
| (∑ i, ρ (closedCell K y c i)) = ρ K := by |
| have h := measure_iUnion₀ (pairwise_aedisjoint_closedCell hρ K y c hy) |
| (fun i => (isClosed_closedCell hK y c i).measurableSet.nullMeasurableSet) |
| simpa only [iUnion_closedCell, tsum_fintype] using h.symm |
|
|
| /-- Integrable functions can be summed cellwise despite the shared boundary faces. -/ |
| theorem sum_integral_closedCell [Fintype ι] [Nonempty ι] |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {ρ : Measure (E d)} (hρ : ρ ≪ volume) {K : Set (E d)} (hK : IsClosed K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| {f : E d → V} (hf : IntegrableOn f K ρ) : |
| (∑ i, ∫ x in closedCell K y c i, f x ∂ρ) = ∫ x in K, f x ∂ρ := by |
| have hi : IntegrableOn f (⋃ i, closedCell K y c i) ρ := by |
| simpa only [iUnion_closedCell] using hf |
| have h := integral_iUnion_ae |
| (fun i => (isClosed_closedCell hK y c i).measurableSet.nullMeasurableSet) |
| (pairwise_aedisjoint_closedCell hρ K y c hy) hi |
| simpa only [iUnion_closedCell, tsum_fintype] using h.symm |
|
|
| /-- Real-valued cell masses also add to the total source mass. -/ |
| theorem sum_real_measure_closedCell [Fintype ι] [Nonempty ι] |
| {ρ : Measure (E d)} [IsFiniteMeasure ρ] (hρ : ρ ≪ volume) |
| {K : Set (E d)} (hK : IsClosed K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) : |
| (∑ i, ρ.real (closedCell K y c i)) = ρ.real K := by |
| have h := sum_integral_closedCell hρ hK y c hy |
| (f := fun _ => (1 : ℝ)) (integrable_const 1) |
| simpa only [integral_const, Measure.real, Measure.restrict_apply_univ, |
| smul_eq_mul, mul_one] using h |
|
|
| /-- Injective sites discharge the null-tie premise in strict activation. -/ |
| theorem exists_strict_active_interior_of_injective [Countable ι] |
| {K : Set (E d)} (hK : Convex ℝ K) (y : ι → E d) (c : ι → ℝ) |
| (hy : Function.Injective y) (i : ι) (hpos : volume (closedCell K y c i) ≠ 0) : |
| ∃ x ∈ interior K, ∀ j, j ≠ i → |
| inner ℝ x (y j) + c j < inner ℝ x (y i) + c i := by |
| apply exists_strict_active_interior hK y c i hpos |
| filter_upwards [ae_affine_tie_implies_slopes_eq volume y c] with x hx |
| intro j hji heq |
| exact hji (hy (hx j i heq)) |
|
|
| /-- Positive cell mass for an absolutely continuous source already guarantees a |
| strictly active interior point. -/ |
| theorem exists_strict_active_interior_of_mass [Countable ι] |
| {ρ : Measure (E d)} (hρ : ρ ≪ volume) {K : Set (E d)} (hK : Convex ℝ K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| (i : ι) (hpos : ρ (closedCell K y c i) ≠ 0) : |
| ∃ x ∈ interior K, ∀ j, j ≠ i → |
| inner ℝ x (y j) + c j < inner ℝ x (y i) + c i := by |
| apply exists_strict_active_interior_of_injective hK y c hy i |
| exact fun hnull => hpos (hρ hnull) |
|
|
| /-- Every equality plane between positive-mass cells cuts the interior of the source. -/ |
| theorem exists_equality_interior_of_positive_cells [Countable ι] |
| {ρ : Measure (E d)} (hρ : ρ ≪ volume) {K : Set (E d)} (hK : Convex ℝ K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| {i j : ι} (hij : i ≠ j) |
| (hi : ρ (closedCell K y c i) ≠ 0) (hj : ρ (closedCell K y c j) ≠ 0) : |
| ∃ x ∈ interior K, inner ℝ x (y i) + c i = inner ℝ x (y j) + c j := by |
| obtain ⟨p, hp, hpi⟩ := exists_strict_active_interior_of_mass hρ hK y c hy i hi |
| obtain ⟨q, hq, hqj⟩ := exists_strict_active_interior_of_mass hρ hK y c hy j hj |
| exact exists_affine_equality_interior hK hq hp (hqj i hij) (hpi j hij.symm) |
|
|
| end CellGeometry |
| end Problem358 |
|
|
| end |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| namespace Problem358.CellGeometry |
|
|
| variable {d : ℕ} {ι : Type*} |
|
|
| /-- With distinct slopes, any maximizing index selects the closed affine cell |
| almost everywhere. No measurability of the supplied selector is required. -/ |
| theorem ae_fiber_eq_closedCell [Countable ι] |
| {K : Set (E d)} (hK : MeasurableSet K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| (k : E d → ι) |
| (hk : ∀ᵐ x ∂uniformMeasure K, ∀ j, |
| ⟪x, y j⟫ + c j ≤ ⟪x, y (k x)⟫ + c (k x)) (i : ι) : |
| {x | k x = i} =ᵐ[uniformMeasure K] closedCell K y c i := by |
| filter_upwards [UniformSource.ae_mem hK, |
| uniform_ae_affine_tie_implies_slopes_eq K y c, hk] with x hx ht hk |
| apply propext |
| constructor |
| · intro hi |
| exact ⟨hx, fun j => hi ▸ hk j⟩ |
| · intro hi |
| apply hy |
| exact ht (k x) i (le_antisymm (hi.2 (k x)) (hk i)) |
|
|
| /-- Closed-cell masses are exactly the masses of any maximizing selector. -/ |
| theorem measure_fiber_eq_closedCell [Countable ι] |
| {K : Set (E d)} (hK : MeasurableSet K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| (k : E d → ι) |
| (hk : ∀ᵐ x ∂uniformMeasure K, ∀ j, |
| ⟪x, y j⟫ + c j ≤ ⟪x, y (k x)⟫ + c (k x)) (i : ι) : |
| uniformMeasure K {x | k x = i} = |
| uniformMeasure K (closedCell K y c i) := |
| measure_congr (ae_fiber_eq_closedCell hK y c hy k hk i) |
|
|
| /-- The real-valued mass form used in the moving-potential energy estimate. -/ |
| theorem real_fiber_eq_closedCell [Countable ι] |
| {K : Set (E d)} (hK : MeasurableSet K) |
| (y : ι → E d) (c : ι → ℝ) (hy : Function.Injective y) |
| (k : E d → ι) |
| (hk : ∀ᵐ x ∂uniformMeasure K, ∀ j, |
| ⟪x, y j⟫ + c j ≤ ⟪x, y (k x)⟫ + c (k x)) (i : ι) : |
| (uniformMeasure K).real {x | k x = i} = |
| (uniformMeasure K).real (closedCell K y c i) := |
| congrArg ENNReal.toReal (measure_fiber_eq_closedCell hK y c hy k hk i) |
|
|
| /-- A single globally measurable selector works for all parameters. Its masses |
| agree with closed-cell masses at every parameter with distinct slopes, so no |
| injectivity at endpoint parameters or outside the time interval is imposed. -/ |
| theorem exists_joint_measurable_selector_closedCell_masses |
| {P : Type*} [MeasurableSpace P] [MeasurableSpace ι] |
| [Fintype ι] [Nonempty ι] |
| {K : Set (E d)} (hK : MeasurableSet K) |
| (y : P → ι → E d) (c : P → ι → ℝ) |
| (hy : ∀ i, Measurable (fun p => y p i)) |
| (hc : ∀ i, Measurable (fun p => c p i)) : |
| ∃ k : P × E d → ι, Measurable k ∧ |
| (∀ p x j, ⟪x, y p j⟫ + c p j ≤ |
| ⟪x, y p (k (p, x))⟫ + c p (k (p, x))) ∧ |
| ∀ p, Function.Injective (y p) → ∀ i, |
| (uniformMeasure K).real {x | k (p, x) = i} = |
| (uniformMeasure K).real (closedCell K (y p) (c p) i) := by |
| have hf (i : ι) : Measurable |
| (fun p : P × E d => ⟪p.2, y p.1 i⟫ + c p.1 i) := |
| (measurable_snd.inner ((hy i).comp measurable_fst)).add |
| ((hc i).comp measurable_fst) |
| obtain ⟨k, hk, _, hmax⟩ := exists_measurable_argmax |
| (fun i (p : P × E d) => ⟪p.2, y p.1 i⟫ + c p.1 i) hf |
| refine ⟨k, hk, fun p x j => hmax (p, x) j, ?_⟩ |
| intro p hp i |
| exact real_fiber_eq_closedCell hK (y p) (c p) hp (fun x => k (p, x)) |
| (Filter.Eventually.of_forall fun x => hmax (p, x)) i |
|
|
| /-- Prescribed closed-cell masses on a chosen set of parameters are realized by |
| one jointly measurable selector, even when sites collide elsewhere. -/ |
| theorem exists_joint_measurable_selector_prescribed_masses |
| {P : Type*} [MeasurableSpace P] [MeasurableSpace ι] |
| [Fintype ι] [Nonempty ι] |
| {K : Set (E d)} (hK : MeasurableSet K) |
| (y : P → ι → E d) (c m : P → ι → ℝ) (s : Set P) |
| (hy : ∀ i, Measurable (fun p => y p i)) |
| (hc : ∀ i, Measurable (fun p => c p i)) |
| (hinj : ∀ p ∈ s, Function.Injective (y p)) |
| (hm : ∀ p ∈ s, ∀ i, |
| (uniformMeasure K).real (closedCell K (y p) (c p) i) = m p i) : |
| ∃ k : P × E d → ι, Measurable k ∧ |
| (∀ p x j, ⟪x, y p j⟫ + c p j ≤ |
| ⟪x, y p (k (p, x))⟫ + c p (k (p, x))) ∧ |
| ∀ p ∈ s, ∀ i, (uniformMeasure K).real {x | k (p, x) = i} = m p i := by |
| obtain ⟨k, hk, hmax, hmass⟩ := |
| exists_joint_measurable_selector_closedCell_masses hK y c hy hc |
| exact ⟨k, hk, hmax, fun p hp i => (hmass p (hinj p hp) i).trans (hm p hp i)⟩ |
|
|
| end Problem358.CellGeometry |
|
|
| end |
|
|
| end OAI |
|
|
