| import OAI.MeasureTheory.Falconer.Recursion.BranchValidity |
| import OAI.MeasureTheory.Falconer.Angular.SourceAngularReduction |
| import OAI.MeasureTheory.Falconer.Recursion.SourcePrefixReduction |
| import OAI.MeasureTheory.Falconer.Recursion.SourceSuffixReduction |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open MeasureTheory |
|
|
| namespace AngularStateData |
|
|
| def branchState (d : AngularStateData) : BranchState := .H d.a₀ d.t₀ d.a d.v d.i |
|
|
| theorem branchState_descriptors (d : AngularStateData) : |
| d.branchState.descriptors d.profile d.N d.q = d.retainedList d.a := rfl |
|
|
| theorem successor_branchState (d : AngularStateData) (p : ℕ) (row : CellPath × CellPath) : |
| (d.successor p row).branchState = .H d.a₀ d.t₀ p d.v d.i := rfl |
|
|
| theorem near_branch_transition (d : AngularStateData) (hw : d.a+100*d.q < d.v) |
| (hn : d.a < d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig) |
| (row : CellPath × CellPath) : |
| BranchStep d.profile d.N d.q d.sig d.branchState .near |
| ((d.successor (d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig) row).branchState) := |
| BranchStep.near hw hn |
|
|
| theorem direct_branch_transition (d : AngularStateData) (hw : d.a+100*d.q < d.v) |
| {e : PolicyDescriptor} (he : DirectPolicyWitness d.profile d.N d.q d.a₀ d.t₀ d.a d.v d.sig e) |
| (row : CellPath × CellPath) : |
| BranchStep d.profile d.N d.q d.sig d.branchState .direct |
| ((d.successor (descriptorAnchor e) row).branchState) := BranchStep.direct hw he |
|
|
| end AngularStateData |
|
|
| namespace PositiveDistanceData |
|
|
| def branchState (d : PositiveDistanceData) : BranchState := .D d.a d.t d.i |
| noncomputable def branchProfile (d : PositiveDistanceData) (n : ℕ) : ℝ := d.mass n-(n:ℝ)/d.N |
|
|
| def angularSuccessor (d : PositiveDistanceData) (p v : ℕ) (row : CellPath × CellPath) : AngularStateData := |
| ⟨d.Q,d.ρ,d.mass,d.N,d.q,d.a,d.t,p,v,d.J,d.i,d.lam,d.sig,row.1,row.2⟩ |
|
|
| theorem angularSuccessor_energy (d : PositiveDistanceData) (p v : ℕ) (row : CellPath × CellPath) |
| (k : Fin 2 → PolicyDescriptor → ℕ) (h : Fin 2 → Plane → ℂ) (Ψ : ℝ → ℝ) : |
| (d.angularSuccessor p v row).energy k h Ψ = |
| d.policyCellEnergy (d.prefixRemainingList p) k h p v Ψ row := rfl |
|
|
| theorem angularSuccessor_list_index (d : PositiveDistanceData) (p v : ℕ) (row : CellPath × CellPath) : |
| (d.angularSuccessor p v row).retainedList p = d.prefixRemainingList p ∧ |
| (d.angularSuccessor p v row).i = d.i ∧ |
| (d.angularSuccessor p v row).profile = d.branchProfile := ⟨rfl,rfl,rfl⟩ |
|
|
| theorem suffix_branch_transition (d : PositiveDistanceData) (hw : d.a+100*d.q < d.t) |
| (hs : d.a+d.t < 2*policyDepth d.branchProfile d.N d.q d.a d.t) : |
| BranchStep d.branchProfile d.N d.q d.sig d.branchState .suffix |
| (({d with t := policyDepth d.branchProfile d.N d.q d.a d.t,i := d.i+1} : PositiveDistanceData).branchState) := |
| BranchStep.suffix hw hs |
|
|
| theorem low_branch_transition (d : PositiveDistanceData) (hw : d.a+100*d.q < d.t) |
| {v : ℕ} (hv : v ∈ d.prefixLowDepths) : |
| BranchStep d.branchProfile d.N d.q d.sig d.branchState .low |
| (({d with t := v,i := d.i+1} : PositiveDistanceData).branchState) := by |
| obtain ⟨hm,hg⟩ := Finset.mem_filter.mp hv |
| exact BranchStep.low hw hm (by linarith) |
|
|
| theorem high_branch_transition (d : PositiveDistanceData) (hw : d.a+100*d.q < d.t) |
| (hs : 2*policyDepth d.branchProfile d.N d.q d.a d.t ≤ d.a+d.t) |
| {v : ℕ} (hv : v ∈ Finset.Ioc d.a d.t) (hg : (d.t:ℝ)-d.sig*d.N < (v:ℝ)) |
| (row : CellPath × CellPath) : |
| BranchStep d.branchProfile d.N d.q d.sig d.branchState .high |
| ((d.angularSuccessor (policyDepth d.branchProfile d.N d.q d.a d.t) v row).branchState) := |
| BranchStep.high hw hs hv (by change (d.t:ℝ)-v < d.sig*d.N; linarith) |
|
|
| theorem high_energy_member_transition (d : PositiveDistanceData) (K : ℕ) (Ψ : ℝ → ℝ) |
| (hw : d.a+100*d.q < d.t) |
| (hs : 2*policyDepth d.branchProfile d.N d.q d.a d.t ≤ d.a+d.t) |
| {E : ℝ} (hE : E ∈ d.prefixSuccessorEnergies K (policyDepth d.branchProfile d.N d.q d.a d.t) Ψ) : |
| E = 0 ∨ ∃ v ∈ Finset.Ioc d.a d.t, ∃ row ∈ sourceProductVertices d.Q d.Q d.ρ d.ρ d.X d.Y |
| (policyDepth d.branchProfile d.N d.q d.a d.t-d.a), |
| ∃ k : Fin 2 → PolicyDescriptor → ℕ, ∃ h : Fin 2 → Plane → ℂ, |
| (∀ l, ∑ e ∈ d.prefixRemainingList (policyDepth d.branchProfile d.N d.q d.a d.t), k l e ≤ K) ∧ |
| (∀ l, Measurable (h l)) ∧ (∀ l x, ‖h l x‖ ≤ 1) ∧ |
| BranchStep d.branchProfile d.N d.q d.sig d.branchState .high |
| ((d.angularSuccessor (policyDepth d.branchProfile d.N d.q d.a d.t) v row).branchState) ∧ |
| E = ((d.angularSuccessor (policyDepth d.branchProfile d.N d.q d.a d.t) v row).energy k h Ψ).toReal := by |
| rcases hE with rfl | ⟨v,hv,hg,row,hr,k,hk,h,hh,hhb,hE⟩ |
| · exact Or.inl rfl |
| · exact Or.inr ⟨v,hv,row,hr,k,h,hk,hh,hhb,d.high_branch_transition hw hs hv hg row,hE⟩ |
|
|
| end PositiveDistanceData |
|
|
| namespace AngularStateData |
|
|
| theorem far_branch_transition (d : AngularStateData) (hw : d.a+100*d.q < d.v) |
| (hn : d.a < d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig) |
| {b : ℕ} (hb : b ∈ Finset.Icc (d.a+9) |
| (d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig+5)) : |
| BranchStep d.profile d.N d.q d.sig d.branchState .far (d.distanceSuccessor b).branchState := |
| BranchStep.far hw hn (by |
| change b ∈ Finset.Icc (d.a+9) |
| (d.v-2*sourceFarBuffer d.N ((d.retainedList d.a).sup descriptorLength) d.sig+6) |
| exact Finset.mem_Icc.mpr |
| ⟨(Finset.mem_Icc.mp hb).1, by have := (Finset.mem_Icc.mp hb).2; omega⟩) |
|
|
| theorem far_successor_list_index (d : AngularStateData) (b : ℕ) : |
| (d.distanceSuccessor b).branchState.descriptors d.profile d.N d.q = |
| policyDescriptorList d.profile d.N d.q b d.v ∧ |
| (d.distanceSuccessor b).i = d.i+1 ∧ |
| (d.distanceSuccessor b).branchProfile = d.profile := ⟨rfl,rfl,rfl⟩ |
|
|
| end AngularStateData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
