openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /RecursiveEnvironment.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Estimates.ActualFarBuffer | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.DirectPolicyWitness | |
| import OAI.MeasureTheory.Falconer.Recursion.Ledger | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.StationaryInverseUnit3 | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphCapScaleInvCrude | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.LiteralDyadicPaths | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale2 | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.TreeCellMembershipConstantOnFinerGrid | |
| 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 | |
| open scoped ContDiff Classical | |
| variable {n : ℕ} | |
| structure RecursiveEnvironmentND (n : ℕ) where | |
| ρ : Measure (EuclideanSpace ℝ (Fin (n+1))) | |
| mass : ℝ → ℝ | |
| N : ℕ | |
| q : ℕ | |
| J : ℕ | |
| sig : ℝ | |
| C : ℝ | |
| namespace RecursiveEnvironmentND | |
| noncomputable def profile (e : RecursiveEnvironmentND n) : ℕ → ℝ := | |
| adjustedProfile (d:=n+1) e.mass e.N | |
| def distanceData (e : RecursiveEnvironmentND n) (a t i : ℕ) (X Y : CellPath (n+1)) : | |
| PositiveDistanceState (n+1) := | |
| ⟨e.ρ,e.mass,e.N,e.q,a,t,e.J,i,e.sig,e.C,X,Y⟩ | |
| end RecursiveEnvironmentND | |
| structure RecursiveAngularND (n : ℕ) extends RecursiveEnvironmentND n where | |
| a₀ : ℕ | |
| t₀ : ℕ | |
| a : ℕ | |
| v : ℕ | |
| i : ℕ | |
| X : CellPath (n+1) | |
| Y : CellPath (n+1) | |
| namespace RecursiveAngularND | |
| noncomputable def profile (e : RecursiveAngularND n) : ℕ → ℝ := e.toRecursiveEnvironmentND.profile | |
| noncomputable def fullList (e : RecursiveAngularND n) : Finset PolicyDescriptor := | |
| policyDescriptorList e.profile e.N e.q e.a₀ e.t₀ | |
| noncomputable def retainedList (e : RecursiveAngularND n) (r : ℕ) : Finset PolicyDescriptor := | |
| unreachedDescriptorList e.profile e.N e.q e.a₀ e.t₀ r | |
| noncomputable def symbol (e : RecursiveAngularND n) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) (r : ℕ) : | |
| EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ := | |
| fun x => (actualResidualSourceSymbol e.profile e.N e.q e.a₀ e.t₀ r L e.ρ e.mass | |
| e.sig e.C e.J e.i f Q x).1 | |
| noncomputable def energy (e : RecursiveAngularND 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) : ℝ := | |
| _root_.OAI.Attempt4Near.sourceAngularHND (n+1) | |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) e.a e.v Ψ | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell e.X) | |
| (((2:ℝ)^e.N)^(-e.mass e.a)) (e.symbol L₁ f Q e.a) z.1 z.2) | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell e.Y) | |
| (((2:ℝ)^e.N)^(-e.mass e.a)) (e.symbol L₂ g P e.a) z.1 z.2) | |
| noncomputable def sameSymbolChildren (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) : ℝ := | |
| _root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax e.ρ e.ρ e.a p e.v e.X e.Y | |
| (((2:ℝ)^e.N)^(-e.mass p)) (((2:ℝ)^e.N)^(-e.mass p)) Ψ | |
| (e.symbol L₁ f Q e.a) (e.symbol L₂ g P e.a) | |
| noncomputable def graphFactor (e : RecursiveAngularND n) (p : ℕ) : ℝ := | |
| ((2:ℝ)^e.N)^(-2*(e.profile p-e.profile e.a)+height e.profile e.a p+ | |
| (e.C+36*(n+1:ℝ)+5)*e.sig) | |
| lemma symbol_measurable (e : RecursiveAngularND n) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (hf : Measurable f) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) (r : ℕ) : | |
| Measurable (Function.uncurry (e.symbol L f Q r)) := | |
| actualResidualSourceSymbol_measurable _ _ _ _ _ _ _ _ _ _ _ _ _ _ hf _ | |
| lemma symbol_smooth (e : RecursiveAngularND n) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) (r : ℕ) | |
| (x : EuclideanSpace ℝ (Fin (n+1))) : ContDiff ℝ ∞ (e.symbol L f Q r x) := | |
| (actualResidualSourceSymbol e.profile e.N e.q e.a₀ e.t₀ r L e.ρ e.mass | |
| e.sig e.C e.J e.i f Q x).2 | |
| end RecursiveAngularND | |
| theorem uniform_recursive_symbol_jets | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (M B total : ℕ) (hW : ∀ L∈W, L.length≤B) {sig : ℝ} (hsig : 0<sig) | |
| (Cq : ℝ) (hCq : 0≤Cq) : | |
| ∃ E : ℝ, 0≤E ∧ ∀ᶠ N : ℕ in atTop, ∀ e : RecursiveAngularND n, | |
| e.N=N → e.sig=sig → 0<e.q → | |
| ∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))), | |
| Measurable f → (∀ x, ‖f x‖≤1) → (∀ b∈e.fullList,L b∈W) → | |
| (∑ b∈e.fullList,(L b).length)≤total → | |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → | |
| _root_.OAI.CurrentFalconer.GraphSymbolJet (e.symbol L f Q e.a) M | |
| (((2:ℝ)^N)^sig*max 1 ((2:ℝ)^((e.retainedList e.a).sup descriptorLength)*((2:ℝ)^N)^(-50*sig))) E := by | |
| obtain ⟨E,hE,h⟩ := uniform_actualResidualSourceSymbol_derivatives W M B total hW sig hsig Cq hCq | |
| refine ⟨E,hE,?_⟩ | |
| filter_upwards [h] with N hN | |
| intro e he hs hq L f Q hf hfn hL htotal hQ | |
| subst N | |
| refine ⟨e.symbol_measurable L f hf Q e.a,e.symbol_smooth L f Q e.a,hE,?_⟩ | |
| intro x j hj w hw | |
| have hh := hN e.profile e.q e.a₀ e.t₀ e.a hq L e.ρ e.mass e.C e.J e.i f Q hL htotal hfn hQ x j hj w hw | |
| simpa only [RecursiveAngularND.symbol,RecursiveAngularND.retainedList,hs,sourceMaskScaleND, | |
| Real.rpow_natCast] using hh | |
| lemma uniform_recursive_symbol_imposes {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ e : RecursiveAngularND n, N₀≤e.N → e.sig=sig → e.J=J → | |
| IsFiniteMeasure e.ρ → 0<e.q → e.t₀≤e.N → | |
| ∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (g p : ℕ), PolicyDescriptor.parallel g p∈e.retainedList e.a → | |
| _root_.OAI.CurrentFalconer.TreeND.ImposesParallelTest e.ρ g p | |
| (_root_.OAI.CurrentFalconer.TreeND.testWidth e.N J (e.i+1) p sig) | |
| (((2:ℝ)^e.N)^(height e.profile g p+e.C*sig)) (e.symbol L f Q e.a) := by | |
| obtain ⟨N₀,hs⟩ := uniform_actualResidualSourceSymbol_support (n:=n) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro e hN he hJe hf hq ht L f Q g p hm P hP x hx w hw hn | |
| have hsupp : w∈tsupport (iteratedFDeriv ℝ 0 | |
| (actualResidualSourceSymbol e.profile e.N e.q e.a₀ e.t₀ e.a L e.ρ e.mass | |
| sig e.C J e.i f Q x).1) := by | |
| apply subset_closure | |
| rw [Function.mem_support,←norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] | |
| simpa only [RecursiveAngularND.symbol,he,hJe] using hn | |
| have hh := hs e.N hN e.ρ hf e.profile e.q e.a₀ e.t₀ e.a hq ht e.mass e.C e.i L f Q 0 x hsupp | |
| have hg := (hh.resolve_left (not_not.mpr hw)) (.parallel g p) hm P | |
| (by simpa only [descriptorAnchor,_root_.OAI.CurrentFalconer.TreeND.literal_dyadicPaths] using hP) | |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hx) | |
| simpa only [descriptorGood,testWidth,_root_.OAI.CurrentFalconer.TreeND.testWidth, | |
| _root_.OAI.CurrentFalconer.TreeND.literal_fullParallelCount,RecursiveAngularND.profile, | |
| RecursiveEnvironmentND.profile] using hg.2 | |
| theorem uniform_recursive_direct_graph {sig : ℝ} (hsig : 0<sig) | |
| (T : ℝ) (J : ℕ) (hJ : 0<J) (Ψ : ℝ → ℝ) (hΨ : ContDiff ℝ ∞ Ψ) | |
| (hsupp : tsupport Ψ⊆Ioo (1/4:ℝ) 4) (hΨ0 : ∀ r,0≤Ψ r) : | |
| ∃ M : ℕ, ∀ (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (B total : ℕ), (∀ L∈W,L.length≤B) → ∀ Cq : ℝ,0≤Cq → | |
| ∃ E : ℝ,0≤E ∧ ∃ N₀ : ℕ, ∀ e : RecursiveAngularND n, | |
| N₀≤e.N → e.sig=sig → e.J=J → e.i<e.J → | |
| ∀ B₀ s,RegularizedProfileData e.ρ e.N B₀ s sig e.mass → | |
| 0<e.q → ⌈10*sig*(e.N:ℝ)⌉₊<e.q → sig*(e.N:ℝ)≤e.q → 10*sig*(e.N:ℝ)≤e.q → | |
| e.a₀≤e.a → e.v≤e.t₀ → e.t₀≤e.N → (e.t₀:ℝ)≤(e.v:ℝ)+sig*e.N → e.a+100*e.q<e.v → | |
| e.v-2*sourceFarBuffer e.N ((e.retainedList e.a).sup descriptorLength) sig≤e.a → | |
| e.X∈dyadicPaths e.a → e.Y∈dyadicPaths e.a → 0<e.ρ (treeCell e.X) → 0<e.ρ (treeCell e.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) (Q P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))), | |
| Measurable f → Measurable g → (∀ x,‖f x‖≤1) → (∀ x,‖g x‖≤1) → | |
| (∀ b∈e.fullList,L₁ b∈W) → (∀ b∈e.fullList,L₂ b∈W) → | |
| (∑ b∈e.fullList,(L₁ b).length)≤total → (∑ b∈e.fullList,(L₂ b).length)≤total → | |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → | |
| (∀ j≤M,∀ w,‖w‖=1 → ‖iteratedFDeriv ℝ j P.1 w‖≤Cq) → | |
| ∃ b : PolicyDescriptor, DirectPolicyWitness e.profile e.N e.q e.a₀ e.t₀ e.a e.v sig b ∧ | |
| e.energy L₁ L₂ f g Q P Ψ ≤ e.graphFactor (descriptorAnchor b)* | |
| e.sameSymbolChildren (descriptorAnchor b) L₁ L₂ f g Q P Ψ+E*((2:ℝ)^e.N)^(-T) := by | |
| let c₀ : ℝ := 1/(4*(n+2:ℝ)) | |
| have hc₀ : 0<c₀ := by dsimp [c₀]; positivity | |
| have hcsmall : ((n+1:ℕ)+1:ℝ)*c₀≤1/2 := by | |
| have he : ((n+1:ℕ)+1:ℝ)*c₀=1/4 := by dsimp [c₀]; push_cast; field_simp; ring | |
| rw [he]; norm_num | |
| obtain ⟨M,C,hC,Ng,hgraph⟩ := _root_.OAI.CurrentFalconer.TreeND.graph_H_finite_jet_estimate | |
| (d:=n+1) hsig hc₀ hcsmall T | |
| obtain ⟨K,hK,hint,hder⟩ := _root_.OAI.CurrentFalconer.graph_fixed_radial_budget hΨ | |
| (IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport Ψ) (hsupp.trans Ioo_subset_Icc_self)) M | |
| obtain ⟨Nt,htest⟩ := uniform_recursive_symbol_imposes (n:=n) hsig hJ | |
| refine ⟨M,?_⟩ | |
| intro W B total hW Cq hCq | |
| obtain ⟨D,hD,hjets⟩ := uniform_recursive_symbol_jets W M B total hW hsig Cq hCq | |
| obtain ⟨Nj,hjets⟩ := Filter.eventually_atTop.mp hjets | |
| refine ⟨C*K*D^4,by positivity,max Ng (max Nt Nj),?_⟩ | |
| intro e hN he hJe hi B₀ s hρ hq hceil hsigN hsig10 ha₀ hv₀ ht₀ ht hw hdir hX hY hoccX hoccY | |
| L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ htotal₁ htotal₂ hQ hP | |
| let := hρ.probability | |
| obtain ⟨b,hb⟩ := actual_direct_policy_witness e.profile hsig.le hq hceil hsigN hsig10 ht hw hdir | |
| refine ⟨b,hb,?_⟩ | |
| let p := descriptorAnchor b | |
| let g₀ := max e.a (p-descriptorLength b) | |
| have hscale : (((e.retainedList e.a).sup descriptorLength : ℕ) : ℝ)=descriptorLength b := by | |
| exact_mod_cast hb.maximal.symm | |
| have hlen : (((e.retainedList e.a).sup descriptorLength : ℕ) : ℝ)≤(e.v:ℝ)-p+2*sig*e.N := by | |
| rw [hscale]; exact hb.length_scale | |
| have hdep : (p:ℝ)-e.a≤(e.v:ℝ)-p+2*sig*e.N := by | |
| have hh := hb.depth_scale | |
| rw [Nat.cast_sub hb.anchor_gt.le] at hh | |
| exact hh | |
| have hgnear : (g₀:ℝ)≤(e.a:ℝ)+2*sig*e.N := by | |
| have hh := hb.g_slack | |
| change (g₀:ℝ)-(e.a:ℝ) ≤ sig*e.N at hh | |
| have : 0 ≤ sig*(e.N:ℝ) := by positivity | |
| linarith | |
| have hj₁ := hjets e.N (by omega) e rfl he hq L₁ f Q hf hfn hL₁ htotal₁ hQ | |
| have hj₂ := hjets e.N (by omega) e rfl he hq L₂ g P hg hgn hL₂ htotal₂ hP | |
| have ht₁ := htest e (by omega) he hJe inferInstance hq ht₀ L₁ f Q g₀ p hb.companion | |
| have ht₂ := htest e (by omega) he hJe inferInstance hq ht₀ L₂ g P g₀ p hb.companion | |
| have hρ' := (_root_.OAI.CurrentFalconer.TreeND.literal_regularizedProfileData_iff e.ρ e.N B₀ s sig e.mass).mp hρ | |
| have hh := hgraph e.N (by omega) e.ρ e.ρ B₀ s s e.mass e.mass hρ' hρ' e.a p (e.v:ℝ) | |
| (((e.retainedList e.a).sup descriptorLength : ℕ) : ℝ) e.C K D J (e.i+1) (e.i+1) g₀ g₀ e.X e.Y Ψ | |
| (e.symbol L₁ f Q e.a) (e.symbol L₂ g P e.a) hb.anchor_gt | |
| (by exact_mod_cast hb.anchor_lt.le) (by exact_mod_cast hv₀.trans ht₀) hdep hlen hb.frequency_slack | |
| (le_max_left _ _) (le_max_left _ _) hb.g_lt hb.g_lt hgnear hgnear hJ (by omega) (by omega) | |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_dyadicPaths] using hX) | |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_dyadicPaths] using hY) | |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hoccX) | |
| (by simpa only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] using hoccY) | |
| hΨ (hsupp.trans Ioo_subset_Icc_self) hΨ0 hint (hder M le_rfl) hK (by simpa only [Real.rpow_natCast] using hj₁) (by simpa only [Real.rpow_natCast] using hj₂) | |
| (by convert ht₁ using 1; rfl) (by convert ht₂ using 1; rfl) | |
| have hfactor : ((2:ℝ)^e.N)^(-((_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N p- | |
| _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.a)+ | |
| (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N p- | |
| _root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N e.a))+ | |
| min (_root_.OAI.PlanarFalconer.height (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) e.a p) | |
| (_root_.OAI.PlanarFalconer.height (_root_.OAI.CurrentFalconer.TreeND.adjustedProfile (d:=n+1) e.mass e.N) e.a p)+ | |
| (e.C+36*(n+1:ℝ)+5)*sig)=e.graphFactor p := by | |
| unfold RecursiveAngularND.graphFactor | |
| rw [he,min_self] | |
| congr 1 | |
| change -((e.profile p-e.profile e.a)+(e.profile p-e.profile e.a))+height e.profile e.a p+_=_ | |
| ring | |
| simp only [Nat.cast_add,Nat.cast_one] at hh | |
| rw [hfactor] at hh | |
| simpa only [RecursiveAngularND.energy,RecursiveAngularND.sameSymbolChildren, | |
| _root_.OAI.CurrentFalconer.TreeND.literal_treeCell,Nat.cast_add,Nat.cast_one] using hh | |
| namespace RecursiveAngularND | |
| noncomputable def reachedChildren (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) : ℝ := | |
| _root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax e.ρ e.ρ e.a p e.v e.X e.Y | |
| (((2:ℝ)^e.N)^(-e.mass p)) (((2:ℝ)^e.N)^(-e.mass p)) Ψ | |
| (e.symbol L₁ f Q p) (e.symbol L₂ g P p) | |
| def successor (e : RecursiveAngularND n) (p : ℕ) (X Y : CellPath (n+1)) : RecursiveAngularND n := | |
| {e with a:=p,X:=X,Y:=Y} | |
| end RecursiveAngularND | |
| theorem uniform_recursive_reached_removal | |
| (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) (Cq : ℝ) (hCq : 0≤Cq) : | |
| ∃ D : ℝ,1≤D ∧ ∀ᶠ N : ℕ in atTop, ∀ e : RecursiveAngularND n, | |
| e.N=N → e.sig=sig → IsFiniteMeasure e.ρ → 0<e.q → ∀ p : ℕ,e.a≤p → | |
| ∀ (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)))), | |
| Measurable f → Measurable g → (∀ x,‖f x‖≤1) → (∀ x,‖g x‖≤1) → | |
| (∀ b∈e.fullList,L₁ b∈W) → (∀ b∈e.fullList,L₂ b∈W) → | |
| (∑ b∈e.fullList,(L₁ b).length)≤total → (∑ b∈e.fullList,(L₂ b).length)≤total → | |
| (∀ w,‖w‖=1 → ‖Q.1 w‖≤Cq) → (∀ w,‖w‖=1 → ‖P.1 w‖≤Cq) → | |
| ∀ Ψ : ℝ → ℝ, Integrable (fun r => Ψ (r/(2:ℝ)^e.v)) → (∀ r,0≤Ψ r) → | |
| e.sameSymbolChildren p L₁ L₂ f g Q P Ψ≤D^(4*total)*e.reachedChildren p L₁ L₂ f g Q P Ψ := by | |
| obtain ⟨D,hD,hrem⟩ := actualResidual_graph_reached_removalND W B total hW hsig Cq hCq | |
| refine ⟨D,hD,?_⟩ | |
| filter_upwards [hrem] with N hrem | |
| intro e he hs hfin hq p hap L₁ L₂ f g Q P hf hg hfn hgn hL₁ hL₂ htotal₁ htotal₂ hQ hP Ψ hiΨ hΨ | |
| subst N | |
| let := hfin | |
| have hcell (X Y : CellPath (n+1)) (hX : X.length=p) (hY : Y.length=p) : | |
| _root_.OAI.Attempt4Near.sourceAngularHND (n+1) | |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) p e.v Ψ | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell X) | |
| (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₁ f Q e.a) z.1 z.2) | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell Y) | |
| (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₂ g P e.a) z.1 z.2) ≤ | |
| D^(4*total)*(e.successor p X Y).energy L₁ L₂ f g Q P Ψ := by | |
| let X' : Fin (n+1) → ℤ := fun j => (pathIndex X j:ℤ) | |
| let Y' : Fin (n+1) → ℤ := fun j => (pathIndex Y j:ℤ) | |
| have hXY : (({X'}:Finset (Fin (n+1) → ℤ)).product {Y'}).Nonempty := by simp | |
| have hh := hrem e.ρ e.ρ e.profile e.q e.a₀ e.t₀ e.a p hq hap L₁ L₂ e.mass e.C e.J e.i | |
| f g Q P hf hg hfn hgn hL₁ hL₂ htotal₁ htotal₂ hQ hP {X'} {Y'} hXY | |
| (((2:ℝ)^e.N)^(-e.mass p)) (((2:ℝ)^e.N)^(-e.mass p)) e.v Ψ (by simpa only [Real.rpow_natCast] using hiΨ) hΨ | |
| have hprod : ({X'} : Finset (Fin (n+1) → ℤ)).product {Y'} = {(X',Y')} := by ext z; simp | |
| simp only [Attempt4Near.graphChildMaxND,hprod,Finset.sup'_singleton] at hh | |
| have heX : treeCell X=Attempt4FullFar.ambientGridCell p X' := by | |
| unfold treeCell; rw [hX]; rfl | |
| have heY : treeCell Y=Attempt4FullFar.ambientGridCell p Y' := by | |
| unfold treeCell; rw [hY]; rfl | |
| change _ ≤ D^(4*total)* | |
| _root_.OAI.Attempt4Near.sourceAngularHND (n+1) | |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) p e.v Ψ | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell X) | |
| (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₁ f Q p) z.1 z.2) | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell Y) | |
| (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₂ g P p) z.1 z.2) | |
| rw [←hs] at hh | |
| rw [heX,heY] | |
| convert hh using 1 <;> rfl | |
| unfold RecursiveAngularND.sameSymbolChildren RecursiveAngularND.reachedChildren | |
| _root_.OAI.CurrentFalconer.TreeND.graphSourceChildMax | |
| split_ifs with hAB | |
| · apply Finset.sup'_le | |
| intro z hz | |
| have hX : z.1.length=p := _root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths p z.1 |>.mp | |
| (Finset.mem_filter.mp (Finset.mem_filter.mp (Finset.mem_product.mp hz).1).1).1 | |
| have hY : z.2.length=p := _root_.OAI.CurrentFalconer.TreeND.mem_dyadicPaths p z.2 |>.mp | |
| (Finset.mem_filter.mp (Finset.mem_filter.mp (Finset.mem_product.mp hz).2).1).1 | |
| have hh := hcell z.1 z.2 hX hY | |
| simp only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] at hh | |
| apply hh.trans | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| change _ ≤ (( _root_.OAI.CurrentFalconer.TreeND.graphChildren e.ρ e.a p e.X).product | |
| (_root_.OAI.CurrentFalconer.TreeND.graphChildren e.ρ e.a p e.Y)).sup' hAB _ | |
| dsimp only [RecursiveAngularND.energy,RecursiveAngularND.successor,RecursiveAngularND.symbol] | |
| simp only [_root_.OAI.CurrentFalconer.TreeND.literal_treeCell] | |
| convert (Finset.le_sup' (f:=fun z => | |
| _root_.OAI.Attempt4Near.sourceAngularHND (n+1) | |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) p e.v Ψ | |
| (fun q => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ | |
| (_root_.OAI.CurrentFalconer.TreeND.treeCell z.1) (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₁ f Q p) q.1 q.2) | |
| (fun q => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ | |
| (_root_.OAI.CurrentFalconer.TreeND.treeCell z.2) (((2:ℝ)^e.N)^(-e.mass p)) (e.symbol L₂ g P p) q.1 q.2)) hz) using 1; rfl | |
| · simp | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |