| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PolicyDescriptorAnchorGtStart |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixDepth |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveEnvironment |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.ResidualAngularTerminal |
| import OAI.MeasureTheory.FalconerCompletion.Shells.PrefixParallelMem |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| open Set MeasureTheory Filter PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar |
| open scoped ContDiff Classical |
| variable {n : ℕ} |
|
|
| noncomputable def recursiveUnit : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))) := |
| ⟨fun _ => 1, by change ContDiff ℝ ∞ (fun _ : EuclideanSpace ℝ (Fin (n+1)) => (1:ℂ)); exact contDiff_const⟩ |
| lemma recursiveUnit_jets (j : ℕ) (w : EuclideanSpace ℝ (Fin (n+1))) : |
| ‖iteratedFDeriv ℝ j (recursiveUnit (n:=n)).1 w‖≤1 := by |
| cases j with |
| | zero => simp [norm_iteratedFDeriv_zero,recursiveUnit] |
| | succ j => simp [recursiveUnit,iteratedFDeriv_succ_const] |
|
|
| namespace RecursiveEnvironmentND |
| noncomputable def angularData (e : RecursiveEnvironmentND n) (a₀ t₀ a v i : ℕ) |
| (X Y : CellPath (n+1)) : RecursiveAngularND n := |
| {e with a₀:=a₀,t₀:=t₀,a:=a,v:=v,i:=i,X:=X,Y:=Y} |
|
|
| noncomputable def energies (e : RecursiveEnvironmentND n) (K : ℝ) |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (total : ℕ) (Ψ : ℝ → ℝ) : BranchState → ℝ → Prop |
| | .D a t i, x => ∃ X Y : CellPath (n+1), X∈dyadicPaths a ∧ Y∈dyadicPaths a ∧ |
| 0<e.ρ (treeCell X) ∧ 0<e.ρ (treeCell Y) ∧ |
| farSeparatedGridND a K (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ)) ∧ |
| x=(e.distanceData a t i X Y).coincidence.toReal |
| | .H a₀ t₀ a v i, x => ∃ X Y : CellPath (n+1), X∈dyadicPaths a ∧ Y∈dyadicPaths a ∧ |
| 0<e.ρ (treeCell X) ∧ 0<e.ρ (treeCell Y) ∧ |
| ∃ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ), |
| Measurable f ∧ Measurable g ∧ (∀ z,‖f z‖≤1) ∧ (∀ z,‖g z‖≤1) ∧ |
| (∀ b∈policyDescriptorList e.profile e.N e.q a₀ t₀,L₁ b∈W) ∧ |
| (∀ b∈policyDescriptorList e.profile e.N e.q a₀ t₀,L₂ b∈W) ∧ |
| (∑ b∈policyDescriptorList e.profile e.N e.q a₀ t₀,(L₁ b).length)≤total ∧ |
| (∑ b∈policyDescriptorList e.profile e.N e.q a₀ t₀,(L₂ b).length)≤total ∧ |
| x=(e.angularData a₀ t₀ a v i X Y).energy L₁ L₂ f g recursiveUnit recursiveUnit Ψ |
| end RecursiveEnvironmentND |
|
|
| lemma RecursiveEnvironmentND.energies_nonneg (e : RecursiveEnvironmentND n) (K : ℝ) |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (total : ℕ) (Ψ : ℝ → ℝ) (hΨ : ∀ r,0≤Ψ r) (s : BranchState) {x : ℝ} |
| (hx : e.energies K W total Ψ s x) : 0≤x := by |
| cases s with |
| | D a t i => obtain ⟨X,Y,hX,hY,hox,hoy,hs,rfl⟩ := hx; exact ENNReal.toReal_nonneg |
| | H a₀ t₀ a v i => |
| obtain ⟨X,Y,hX,hY,hox,hoy,L₁,L₂,f,g,hf,hg,hfn,hgn,hL₁,hL₂,ht₁,ht₂,rfl⟩ := hx |
| exact _root_.OAI.Attempt4Near.sourceAngularHND_nonneg _ _ _ _ hΨ _ _ |
|
|
| lemma unreachedDescriptorList_initial (A : ℕ → ℝ) (N q a t : ℕ) (hq : 0<q) : |
| unreachedDescriptorList A N q a t a=policyDescriptorList A N q a t := by |
| apply Finset.filter_eq_self.mpr |
| intro b hb |
| exact policyDescriptor_anchor_gt_start A hq hb |
|
|
| namespace PositiveDistanceState |
| noncomputable def prefixAngularData (e : PositiveDistanceState (n+1)) (v : ℕ) : RecursiveAngularND n := |
| ⟨⟨e.ρ,e.mass,e.N,e.q,e.J,e.sig,e.C⟩,e.a,e.t,e.a,v,e.i,e.X,e.Y⟩ |
|
|
| lemma prefixPhased_symbol (e : PositiveDistanceState (n+1)) (hq : 0<e.q) (v : ℕ) |
| (z : PrefixAssignment n) (ξ : EuclideanSpace ℝ (Fin (n+1))) : |
| e.prefixPhasedLeft z ξ=(e.prefixAngularData v).symbol (prefixLeftWords z) |
| (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase (n+1) ((2:ℝ)^e.a)⁻¹ ξ) recursiveUnit e.a ∧ |
| e.prefixPhasedRight z ξ=(e.prefixAngularData v).symbol (prefixRightWords z) |
| (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase (n+1) ((2:ℝ)^e.a)⁻¹ ξ) recursiveUnit e.a := by |
| have he := unreachedDescriptorList_initial (adjustedProfile (d:=n+1) e.mass e.N) e.N e.q e.a e.t hq |
| constructor <;> ext x w |
| · simp only [prefixPhasedLeft,prefixInverseLeft_eq,RecursiveAngularND.symbol, |
| actualResidualSourceSymbol,prefixAngularData,RecursiveAngularND.profile, |
| RecursiveEnvironmentND.profile,recursiveUnit,one_mul,he,prefixDescriptors] |
| · simp only [prefixPhasedRight,prefixInverseRight_eq,RecursiveAngularND.symbol, |
| actualResidualSourceSymbol,prefixAngularData,RecursiveAngularND.profile, |
| RecursiveEnvironmentND.profile,recursiveUnit,one_mul,he,prefixDescriptors] |
|
|
| lemma prefixGraphChildMax_sameSymbol (e : PositiveDistanceState (n+1)) (hq : 0<e.q) (v : ℕ) |
| (Ψ : ℝ → ℝ) (z : PrefixAssignment n) (ξ : EuclideanSpace ℝ (Fin (n+1))) : |
| e.prefixGraphChildMax v Ψ z ξ=(e.prefixAngularData v).sameSymbolChildren e.sourcePrefixDepth |
| (prefixLeftWords z) (prefixRightWords z) |
| (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase (n+1) ((2:ℝ)^e.a)⁻¹ ξ) |
| (Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase (n+1) ((2:ℝ)^e.a)⁻¹ ξ) |
| recursiveUnit recursiveUnit Ψ := by |
| unfold prefixGraphChildMax |
| rw [(e.prefixPhased_symbol hq v z ξ).1,(e.prefixPhased_symbol hq v z ξ).2] |
| rfl |
| end PositiveDistanceState |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| open Set MeasureTheory Filter PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar |
| open scoped ContDiff Classical |
| variable {n : ℕ} |
|
|
| lemma RecursiveAngularND.reachedChildren_le (e : RecursiveAngularND n) (p : ℕ) |
| (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) |
| (Q P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) (Ψ : ℝ → ℝ) (M : ℝ) (hM : 0≤M) |
| (hb : ∀ X Y : CellPath (n+1),X∈dyadicPaths p → Y∈dyadicPaths p → |
| 0<e.ρ (treeCell X) → 0<e.ρ (treeCell Y) → |
| (e.successor p X Y).energy L₁ L₂ f g Q P Ψ≤M) : |
| e.reachedChildren p L₁ L₂ f g Q P Ψ≤M := by |
| unfold reachedChildren _root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax |
| split_ifs with hAB |
| · apply Finset.sup'_le |
| intro z hz |
| have hx := Finset.mem_filter.mp (Finset.mem_product.mp hz).1 |
| have hy := Finset.mem_filter.mp (Finset.mem_product.mp hz).2 |
| have hh := hb z.1 z.2 ((mem_dyadicPaths p z.1).mpr (_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths p z.1 |>.mp (Finset.mem_filter.mp hx.1).1)) |
| ((mem_dyadicPaths p z.2).mpr (_root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths p z.2 |>.mp (Finset.mem_filter.mp hy.1).1)) |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hx.2) |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hy.2) |
| simp only [RecursiveAngularND.energy,_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] at hh |
| convert hh using 1; rfl |
| · exact hM |
| |
| lemma RecursiveAngularND.successor_energy_mem (e : RecursiveAngularND n) (K : ℝ) |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (total : ℕ) (Ψ : ℝ → ℝ) (p : ℕ) (X Y : CellPath (n+1)) |
| (hX : X∈dyadicPaths p) (hY : Y∈dyadicPaths p) |
| (hox : 0<e.ρ (treeCell X)) (hoy : 0<e.ρ (treeCell Y)) |
| (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (hf : Measurable f) (hg : Measurable g) |
| (hfn : ∀ x,‖f x‖≤1) (hgn : ∀ x,‖g x‖≤1) |
| (hw₁ : ∀ b∈e.fullList,L₁ b∈W) (hw₂ : ∀ b∈e.fullList,L₂ b∈W) |
| (ht₁ : (∑ b∈e.fullList,(L₁ b).length)≤total) (ht₂ : (∑ b∈e.fullList,(L₂ b).length)≤total) : |
| e.toRecursiveEnvironmentND.energies K W total Ψ (.H e.a₀ e.t₀ p e.v e.i) |
| ((e.successor p X Y).energy L₁ L₂ f g recursiveUnit recursiveUnit Ψ) := |
| ⟨X,Y,hX,hY,hox,hoy,L₁,L₂,f,g,hf,hg,hfn,hgn,hw₁,hw₂,ht₁,ht₂,rfl⟩ |
| |
| theorem uniform_recursive_high_closure |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (B total Order : ℕ) (hW : ∀ L∈W,L.length≤B) |
| (hprefix : prefixInverseVocabulary n Order⊆W) (htotal : 2*Order≤total) |
| {sig : ℝ} (hsig : 0<sig) : |
| ∃ D : ℝ,1≤D ∧ ∃ N₀ : ℕ, ∀ e : PositiveDistanceState (n+1), |
| N₀≤e.N → e.sig=sig → IsFiniteMeasure e.ρ → 0<e.q → e.a≤e.sourcePrefixDepth → |
| ∀ (v : ℕ) (Ψ : ℝ → ℝ), Integrable (fun r => Ψ (r/(2:ℝ)^v)) → (∀ r,0≤Ψ r) → |
| ∀ K M : ℝ,0≤M → |
| (∀ x, (e.prefixAngularData v).toRecursiveEnvironmentND.energies K W total Ψ |
| (.H e.a e.t e.sourcePrefixDepth v e.i) x → x≤M) → |
| e.prefixHighChildSup Order v Ψ≤D^(4*total)*M := by |
| obtain ⟨D,hD,hrem⟩ := uniform_recursive_reached_removal W B total hW hsig 1 (by norm_num) |
| obtain ⟨N₀,hrem⟩ := eventually_atTop.mp hrem |
| refine ⟨D,hD,N₀,?_⟩ |
| intro e hN hs hfin hq hap v Ψ hiΨ hΨ K M hM hbound |
| apply csSup_le (by exact ⟨0,Set.mem_insert _ _⟩) |
| intro r hr |
| rcases hr with rfl | ⟨j,hj,z,hz,ξ,rfl⟩ |
| · positivity |
| let a := e.prefixAngularData v |
| let f := Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase (n+1) ((2:ℝ)^e.a)⁻¹ ξ |
| have hf : Measurable f := by |
| exact ((Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_continuous (n+1) |
| ((2:ℝ)^e.a)⁻¹).comp (continuous_const.prodMk continuous_id)).measurable |
| have hfn : ∀ x,‖f x‖≤1 := fun x => by |
| dsimp only [f]; rw [Attempt4PrefixShell.SpatialFourierND.spatialFourierPhase_norm] |
| have hw₁ : ∀ b∈a.fullList,prefixLeftWords z b∈W := fun b _ => |
| hprefix (prefixInverseVocabulary_mem Order j hj _ hz (Sum.inl b)) |
| have hw₂ : ∀ b∈a.fullList,prefixRightWords z b∈W := fun b _ => |
| hprefix (prefixInverseVocabulary_mem Order j hj _ hz (Sum.inr b)) |
| have htot := e.prefixInverseWord_total j hz |
| have ht₁ : (∑ b∈a.fullList,(prefixLeftWords z b).length)≤total := by |
| exact (Nat.le_add_right _ _).trans (htot.trans ((Nat.mul_le_mul_left 2 hj).trans htotal)) |
| have ht₂ : (∑ b∈a.fullList,(prefixRightWords z b).length)≤total := by |
| exact (Nat.le_add_left _ _).trans (htot.trans ((Nat.mul_le_mul_left 2 hj).trans htotal)) |
| have hb := hrem e.N hN a rfl hs hfin hq e.sourcePrefixDepth hap |
| (prefixLeftWords z) (prefixRightWords z) f f recursiveUnit recursiveUnit hf hf hfn hfn |
| hw₁ hw₂ ht₁ ht₂ (fun _ _ => by norm_num [recursiveUnit]) |
| (fun _ _ => by norm_num [recursiveUnit]) Ψ hiΨ hΨ |
| rw [e.prefixGraphChildMax_sameSymbol hq v Ψ z ξ] |
| apply hb.trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply a.reachedChildren_le _ _ _ _ _ _ _ _ M hM |
| intro X Y hX hY hox hoy |
| exact hbound _ (a.successor_energy_mem K W total Ψ _ X Y hX hY hox hoy |
| (prefixLeftWords z) (prefixRightWords z) f f hf hf hfn hfn hw₁ hw₂ ht₁ ht₂) |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
| |
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND |
| open Set MeasureTheory Filter PlanarFalconer |
| open scoped ContDiff Classical |
| variable {n : ℕ} |
| |
| theorem uniform_recursive_terminal |
| {Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) |
| (B total : ℕ) (hW : ∀ L∈W,L.length≤B) {sig : ℝ} (hsig : 0<sig) : |
| ∃ H : ℝ,1≤H ∧ ∃ N₀ : ℕ,0<N₀ ∧ ∀ e : RecursiveEnvironmentND n, |
| N₀≤e.N → e.sig=sig → 0<e.q → ∀ B₀ t,RegularizedProfileData e.ρ e.N B₀ t sig e.mass → |
| ∀ K : ℝ,∀ s : BranchState,s.Valid e.N sig → s.finish ≤ s.start+100*e.q → |
| ∀ x,e.energies K W total Ψ s x → |
| x≤H*((2:ℝ)^e.N)^(100*(n+1:ℝ)*(e.q:ℝ)/e.N+4*(n+2:ℝ)*sig) := by |
| obtain ⟨H,hH,hbound⟩ := actual_residual_angular_terminal hΨ W B total hW hsig 1 (by norm_num) |
| obtain ⟨N₀,hbound⟩ := eventually_atTop.mp hbound |
| refine ⟨max 1 H,le_max_left _ _,max 1 N₀,by omega,?_⟩ |
| intro e hN hs hq B₀ t hd K s hv ht x hx |
| let := hd.probability |
| have hNp : 0<e.N := by omega |
| have hp : 0≤((2:ℝ)^e.N)^(100*(n+1:ℝ)*(e.q:ℝ)/e.N+4*(n+2:ℝ)*sig) := by positivity |
| cases s with |
| | D a v i => |
| obtain ⟨X,Y,hX,hY,hox,hoy,hsep,rfl⟩ := hx |
| have hb := (e.distanceData a v i X Y).actual_distance_terminal_rpow hd hNp |
| (hv.1.trans hv.2) ((mem_dyadicPaths a X).mp hX) ((mem_dyadicPaths a Y).mp hY) ht |
| apply hb.trans |
| apply le_trans _ (le_mul_of_one_le_left hp (le_max_left _ _)) |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) |
| change 100*(e.q:ℝ)/e.N+4*((n+1:ℕ)+1:ℝ)*sig≤_ |
| push_cast |
| have hn : 0≤(n:ℝ) := Nat.cast_nonneg n |
| have hqn : 100*(e.q:ℝ) ≤ 100*((n:ℝ)+1)*(e.q:ℝ) := by nlinarith [(Nat.cast_nonneg e.q : (0:ℝ)≤(e.q:ℝ))] |
| have hh := div_le_div_of_nonneg_right hqn (Nat.cast_nonneg e.N : (0:ℝ)≤(e.N:ℝ)) |
| linarith |
| | H a₀ t₀ a v i => |
| obtain ⟨X,Y,hX,hY,hox,hoy,L₁,L₂,f,g,hf,hg,hfn,hgn,hw₁,hw₂,ht₁,ht₂,rfl⟩ := hx |
| have htR : (v:ℝ)-(a:ℝ)≤100*(e.q:ℝ) := by |
| have ht' : v≤a+100*e.q := ht |
| exact_mod_cast (show (v:ℤ)-(a:ℤ)≤100*(e.q:ℤ) by omega) |
| have hh := hbound e.N (by omega) e.ρ e.ρ e.profile e.q a₀ t₀ a hq hNp |
| (hv.2.1.trans (hv.2.2.1.trans hv.2.2.2.1)) L₁ L₂ e.mass e.C e.J i f g recursiveUnit recursiveUnit |
| X Y v sig t t B₀ B₀ hd hd ((mem_dyadicPaths a X).mp hX) ((mem_dyadicPaths a Y).mp hY) |
| htR hf hg hfn hgn hw₁ hw₂ ht₁ ht₂ (fun _ _ => by norm_num [recursiveUnit]) |
| (fun _ _ => by norm_num [recursiveUnit]) |
| rw [←hs] at hh |
| change _ ≤ max 1 H * _ |
| have hh' : (e.angularData a₀ t₀ a v i X Y).energy L₁ L₂ f g recursiveUnit recursiveUnit Ψ≤ |
| H*((2:ℝ)^e.N)^(100*(n+1:ℝ)*e.q/e.N+4*(n+2:ℝ)*sig) := by |
| simp only [RecursiveAngularND.energy,_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] |
| simp only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] at hh |
| convert hh using 1 <;> first | rfl | rw [hs] |
| exact hh'.trans (mul_le_mul_of_nonneg_right (le_max_right _ _) hp) |
| end GraphLiteralSupport.CurrentFalconer.TreeND |
| end |
| end |
|
|
| end OAI |
|
|
