openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /PolicyDescriptorAnchorGtStart.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.InteriorPolicy | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.FarChildMaximum | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.ResidualAngular | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set MeasureTheory Filter PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ENNReal Topology Classical | |
| variable {n : ℕ} | |
| lemma policyDescriptor_anchor_gt_start (A : ℕ → ℝ) {N q a t : ℕ} | |
| (hq : 0<q) {b : PolicyDescriptor} (hb : b∈policyDescriptorList A N q a t) : | |
| a<descriptorAnchor b := by | |
| obtain ⟨z,hz,hb⟩ := Finset.mem_biUnion.mp hb | |
| obtain ⟨hz,ha,ht,hwide⟩ := Finset.mem_filter.mp hz | |
| have huN : z.2≤N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz).2) | |
| rw [originDescriptor_anchor A hb] | |
| have hp := (policyDepth_interior A hq (policyBlocks_nonempty hq huN hwide)).1 | |
| omega | |
| lemma policyDescriptorList_subset_unreached | |
| (A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q) | |
| (ha : a₀≤a) (hab : a<b) (ht : t≤t₀) : | |
| policyDescriptorList A N q b t ⊆ unreachedDescriptorList A N q a₀ t₀ a := by | |
| intro e he | |
| exact Finset.mem_filter.mpr ⟨policyDescriptorList_mono A (ha.trans hab.le) ht he, | |
| hab.trans (policyDescriptor_anchor_gt_start A hq he)⟩ | |
| lemma actualResidualAmbientGood_subset_farChild | |
| (A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q) | |
| (ha : a₀≤a) (hab : a<b) (ht : t≤t₀) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) | |
| (sig C : ℝ) (J i : ℕ) (x : EuclideanSpace ℝ (Fin (n+1))) : | |
| actualResidualAmbientGood A N q a₀ t₀ a ρ m sig C J i x ⊆ | |
| actualPolicyAmbientGood A N q b t ρ m sig C J i x := by | |
| intro w hw | |
| rcases hw with hw | hw | |
| · exact Or.inl hw | |
| · exact Or.inr (fun e he => hw e (policyDescriptorList_subset_unreached A hq ha hab ht he)) | |
| lemma actualResidualPairAmbientGood_subset_farChild | |
| (A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q) | |
| (ha : a₀≤a) (hab : a<b) (ht : t≤t₀) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) | |
| (sig C : ℝ) (J i : ℕ) (z : EuclideanSpace ℝ (Fin (n+1))×EuclideanSpace ℝ (Fin (n+1))) : | |
| actualResidualPairAmbientGood A N q a₀ t₀ a ρ m sig C J i z ⊆ | |
| actualPairAmbientGood A N q b t ρ m sig C J i z := by | |
| intro w hw | |
| exact ⟨actualResidualAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i z.1 hw.1, | |
| actualResidualAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i z.2 hw.2⟩ | |
| lemma positiveDistanceCountND_mono | |
| {V : Type*} [PseudoMetricSpace V] [MeasurableSpace V] | |
| (μ ν : Measure (V×V)) [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (hμ : μ≤ν) (d : ℕ) (v a M : ℝ) : | |
| positiveDistanceCountND μ d v a M ≤ positiveDistanceCountND ν d v a M := by | |
| exact mul_le_mul' le_rfl ((Measure.prod_mono hμ hμ) _) | |
| lemma farChildMaximumND_mono | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| (a : ℕ) (P Q : Finset ((Fin (n+1) → ℤ)×(Fin (n+1) → ℤ))) (v M : ℝ) | |
| (U V U' V' : (EuclideanSpace ℝ (Fin (n+1))×EuclideanSpace ℝ (Fin (n+1))) → Set (EuclideanSpace ℝ (Fin (n+1)))) | |
| (hU : ∀ z, U z⊆U' z) (hV : ∀ z, V z⊆V' z) : | |
| farChildMaximumND ρ κ a P Q v M U V ≤ farChildMaximumND ρ κ a P Q v M U' V' := by | |
| apply sup_le_sup | |
| · apply Finset.sup_mono_fun | |
| intro p hp | |
| exact positiveDistanceCountND_mono | |
| ((gridPairMeasureND ρ a p).restrict {z | euclideanPairDirectionND z∈U z}) | |
| ((gridPairMeasureND ρ a p).restrict {z | euclideanPairDirectionND z∈U' z}) | |
| (Measure.restrict_mono (fun z hz => hU z hz) le_rfl) _ _ _ _ | |
| · apply Finset.sup_mono_fun | |
| intro p hp | |
| exact positiveDistanceCountND_mono | |
| ((gridPairMeasureND κ a p).restrict {z | euclideanPairDirectionND z∈V z}) | |
| ((gridPairMeasureND κ a p).restrict {z | euclideanPairDirectionND z∈V' z}) | |
| (Measure.restrict_mono (fun z hz => hV z hz) le_rfl) _ _ _ _ | |
| lemma actualResidual_farChildMaximum_le | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| (A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q) | |
| (ha : a₀≤a) (hab : a<b) (ht : t≤t₀) | |
| (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (P Q : Finset ((Fin (n+1) → ℤ)×(Fin (n+1) → ℤ))) (v M : ℝ) : | |
| farChildMaximumND ρ κ b P Q v M | |
| (actualResidualPairAmbientGood A N q a₀ t₀ a ρ m sig C J i) | |
| (actualResidualPairAmbientGood A N q a₀ t₀ a κ m sig C J i) ≤ | |
| farChildMaximumND ρ κ b P Q v M | |
| (actualPairAmbientGood A N q b t ρ m sig C J i) | |
| (actualPairAmbientGood A N q b t κ m sig C J i) := | |
| farChildMaximumND_mono ρ κ b P Q v M _ _ _ _ | |
| (actualResidualPairAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i) | |
| (actualResidualPairAmbientGood_subset_farChild A hq ha hab ht κ m sig C J i) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |