openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /OccupiedDescendantPath.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.BranchCounts | |
| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.ResidualFullChildNearFar | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.PairAmbientGoodRadialIffDistanceGood | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.Extend | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.ProfileLipschitz | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveEnvironment | |
| 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 ENNReal | |
| variable {n : ℕ} | |
| lemma occupiedDescendant_path (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) | |
| (P : CellPath (n+1)) {a b : ℕ} (hP : P.length=a) (hab : a≤b) | |
| (Z : Fin (n+1) → ℤ) | |
| (hZ : Z∈occupiedDescendantsND ρ a b (fun j => (pathIndex P j:ℤ))) : | |
| ∃ Q : CellPath (n+1), Q∈dyadicPaths b ∧ Q.take a=P ∧ | |
| 0<ρ (treeCell Q) ∧ treeCell Q=Attempt4FullFar.ambientGridCell b Z := by | |
| have hz := Finset.mem_filter.mp hZ | |
| obtain ⟨Q,hQ,hidx,hanc⟩ := exists_treePath_of_ambientDescendant P (hP ▸ hab) Z | |
| (by simpa only [hP] using hz.1) | |
| have he : treeCell Q=Attempt4FullFar.ambientGridCell b Z := by | |
| unfold treeCell; rw [hQ,hidx]; rfl | |
| exact ⟨Q,(mem_dyadicPaths b Q).mpr hQ,by simpa only [hP] using hanc, | |
| by rw [he]; exact pos_iff_ne_zero.mpr hz.2,he⟩ | |
| lemma RegularizedProfileData.occupiedDescendant_lower | |
| {ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))} {N B : ℕ} {s sig : ℝ} {m : ℝ → ℝ} | |
| (hd : RegularizedProfileData ρ N B s sig m) | |
| (P : CellPath (n+1)) {a b : ℕ} (hP : P.length=a) (hab : a≤b) (hb : b≤N) | |
| (Z : Fin (n+1) → ℤ) | |
| (hZ : Z∈occupiedDescendantsND ρ a b (fun j => (pathIndex P j:ℤ))) : | |
| ((2:ℝ)^N)^(-m b-(n+2:ℝ)*sig) ≤ (ρ (Attempt4FullFar.ambientGridCell b Z)).toReal := by | |
| obtain ⟨Q,hQ,hanc,hocc,he⟩ := occupiedDescendant_path ρ P hP hab Z hZ | |
| have hh := (hd.cells b hb Q ((mem_dyadicPaths b Q).mp hQ) hocc).1 | |
| rw [he,←Real.rpow_add (by positivity : (0:ℝ)<2^N)] at hh | |
| convert hh using 1; congr 1; push_cast; ring | |
| lemma recursive_near_max_le (e : RecursiveAngularND n) (h : ℕ) (hah : e.a≤h) | |
| (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| (_hx : e.ρ (treeCell e.X)≠0) (_hy : e.ρ (treeCell e.Y)≠0) | |
| (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) | |
| (hb : ∀ X Y : CellPath (n+1), X∈dyadicPaths h → Y∈dyadicPaths h → | |
| 0<e.ρ (treeCell X) → 0<e.ρ (treeCell Y) → | |
| (e.successor h X Y).energy L₁ L₂ f g Q P Ψ≤M) : | |
| let X := fun j => (pathIndex e.X j:ℤ) | |
| let Y := fun j => (pathIndex e.Y j:ℤ) | |
| let A := occupiedDescendantsND e.ρ e.a h X | |
| let B := occupiedDescendantsND e.ρ e.a h Y | |
| ∀ hAB : (A.product B).Nonempty, | |
| Attempt4Near.graphChildMaxND (n+1) | |
| (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h e.v Ψ A B hAB | |
| (originalCellTransformND e.ρ h (((2:ℝ)^e.N)^(-e.mass h)) (e.symbol L₁ f Q h)) | |
| (originalCellTransformND e.ρ h (((2:ℝ)^e.N)^(-e.mass h)) (e.symbol L₂ g P h))≤M := by | |
| dsimp only | |
| intro hAB | |
| apply Finset.sup'_le | |
| intro z hz | |
| obtain ⟨X,hXp,hXa,hXo,hXg⟩ := occupiedDescendant_path e.ρ e.X hX hah z.1 (Finset.mem_product.mp hz).1 | |
| obtain ⟨Y,hYp,hYa,hYo,hYg⟩ := occupiedDescendant_path e.ρ e.Y hY hah z.2 (Finset.mem_product.mp hz).2 | |
| have hh := hb X Y hXp hYp hXo hYo | |
| change _root_.OAI.Attempt4Near.sourceAngularHND (n+1) | |
| (_root_.OAI.Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h e.v Ψ | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell X) | |
| (((2:ℝ)^e.N)^(-e.mass h)) (e.symbol L₁ f Q h) z.1 z.2) | |
| (fun z => _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell Y) | |
| (((2:ℝ)^e.N)^(-e.mass h)) (e.symbol L₂ g P h) z.1 z.2)≤M at hh | |
| rw [hXg,hYg] at hh | |
| exact hh | |
| lemma recursive_far_max_le (e : RecursiveAngularND n) [IsFiniteMeasure e.ρ] | |
| (b : ℕ) (hab : e.a≤b) (hX : e.X.length=e.a) (hY : e.Y.length=e.a) | |
| {K M : ℝ} (hK : 32*(n+2:ℝ)≤K) (hM : 0≤M) | |
| (hb : ∀ X Y : CellPath (n+1), X∈dyadicPaths b → Y∈dyadicPaths b → | |
| 0<e.ρ (treeCell X) → 0<e.ρ (treeCell Y) → | |
| farSeparatedGridND b K (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ)) → | |
| (e.toRecursiveEnvironmentND.distanceData b e.v (e.i+1) X Y).coincidence.toReal≤M) : | |
| (farChildMaximumND e.ρ e.ρ b | |
| (occupiedFarPairsND e.ρ e.a b K (fun j => (pathIndex e.X j:ℤ))) | |
| (occupiedFarPairsND e.ρ e.a b K (fun j => (pathIndex e.Y j:ℤ))) | |
| e.v (((2:ℝ)^e.N)^(-e.mass b)) | |
| (actualPairAmbientGood e.profile e.N e.q b e.v e.ρ e.mass e.sig e.C e.J (e.i+1)) | |
| (actualPairAmbientGood e.profile e.N e.q b e.v e.ρ e.mass e.sig e.C e.J (e.i+1))).toReal≤M := by | |
| have one (P : CellPath (n+1)) (hP : P.length=e.a) : | |
| (occupiedFarPairsND e.ρ e.a b K (fun j => (pathIndex P j:ℤ))).sup | |
| (fun p => positiveDistanceCountND | |
| ((gridPairMeasureND e.ρ b p).restrict {z | euclideanPairDirectionND z∈ | |
| actualPairAmbientGood e.profile e.N e.q b e.v e.ρ e.mass e.sig e.C e.J (e.i+1) z}) | |
| (n+1) e.v b (((2:ℝ)^e.N)^(-e.mass b)))≤ENNReal.ofReal M := by | |
| apply Finset.sup_le | |
| intro p hp | |
| obtain ⟨X,Y,hX,hY,hXa,hYa,hXp,hYp,hXo,hYo,he⟩ := | |
| occupiedFarPair_actual_state e.ρ e.mass e.N e.q b e.v e.sig e.C e.J (e.i+1) | |
| P (hP ▸ hab) hK p (by simpa only [hP] using hp) | |
| have hsep := (mem_farParentPairsND_iff.mp (Finset.mem_filter.mp hp).1).2.2 | |
| have hs : farSeparatedGridND b K (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ)) := by | |
| rwa [hXp,hYp] | |
| have hh := hb X Y ((mem_dyadicPaths b X).mpr hX) ((mem_dyadicPaths b Y).mpr hY) | |
| (pos_iff_ne_zero.mpr hXo) (pos_iff_ne_zero.mpr hYo) hs | |
| apply (ENNReal.toReal_le_toReal (positiveDistanceCountND_ne_top _ _ _ _ _) | |
| ENNReal.ofReal_ne_top).mp | |
| rw [ENNReal.toReal_ofReal hM] | |
| dsimp only [RecursiveAngularND.profile,RecursiveEnvironmentND.profile] | |
| rw [he] | |
| exact hh | |
| exact (ENNReal.toReal_le_toReal (farChildMaximumND_ne_top _ _ _ _ _ _ _ _ _) | |
| ENNReal.ofReal_ne_top).mpr (sup_le (one e.X hX) (one e.Y hY)) |>.trans (by simp [hM]) | |
| theorem uniform_recursive_near_far | |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hsupp : tsupport Ψ⊆Ioo (1/4:ℝ) 4) | |
| (hΨpos : ∀ t,0≤Ψ t) {sig : ℝ} (hsig : 0<sig) (hsig₁ : sig≤1) | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (B total : ℕ) (hW : ∀ L∈W,L.length≤B) | |
| {Jmask : ℕ} (hJmask : 0<Jmask) (Cq : ℝ) (hCq : 0≤Cq) | |
| (T D : ℝ) (ell : ℕ) (hK : 32*(n+2:ℝ)≤(2:ℝ)^ell) : | |
| ∃ M : ℕ, ∃ Dnear H : ℝ,1≤Dnear ∧ 0≤H ∧ ∃ N₀ : ℕ, ∀ e : RecursiveAngularND n, | |
| N₀≤e.N → e.sig=sig → e.J=Jmask → | |
| ∀ B₀ s,RegularizedProfileData e.ρ e.N B₀ s sig e.mass → | |
| 0<e.q → e.t₀≤e.N → e.a₀≤e.a → e.v≤e.t₀ → | |
| ∀ h : ℕ,e.a<h → h+ell+2≤e.N → | |
| ((h+ell+2:ℕ):ℝ)≤(e.v:ℝ)-2*(max ((e.retainedList e.a).sup descriptorLength) | |
| (Nat.ceil (10*sig*e.N)):ℕ)+D → | |
| 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)))) | |
| (Mn : ℝ) (Mf : ℕ → ℝ), | |
| 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) → | |
| (∀ X Y : CellPath (n+1),X∈dyadicPaths h → Y∈dyadicPaths h → | |
| 0<e.ρ (treeCell X) → 0<e.ρ (treeCell Y) → | |
| (e.successor h X Y).energy L₁ L₂ f g Q P Ψ≤Mn) → | |
| (∀ b∈Finset.Icc (e.a+1) (h+ell+2),0≤Mf b) → | |
| (∀ b∈Finset.Icc (e.a+1) (h+ell+2),∀ X Y : CellPath (n+1), | |
| X∈dyadicPaths b → Y∈dyadicPaths b → 0<e.ρ (treeCell X) → 0<e.ρ (treeCell Y) → | |
| farSeparatedGridND b ((2:ℝ)^ell) (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ)) → | |
| (e.toRecursiveEnvironmentND.distanceData b e.v (e.i+1) X Y).coincidence.toReal≤Mf b) → | |
| e.energy L₁ L₂ f g Q P Ψ≤ | |
| (20*(n+2:ℝ)+5)^(2*(n+1))*Dnear^(4*total)* | |
| ((2:ℝ)^e.N)^(-2*(e.profile h-e.profile e.a)+4*(n+2:ℝ)*sig)*Mn+ | |
| H*(∑ b∈Finset.Icc (e.a+1) (h+ell+2), | |
| ((2:ℝ)^e.N)^(-2*(e.profile b-e.profile e.a)+4*(n+2:ℝ)*sig)*Mf b)+ | |
| ((2:ℝ)^e.N)^(-T) := by | |
| have hs : ∀ t,Ψ t≠0 → t∈Ioo (1/4:ℝ) 4 := | |
| fun t ht => hsupp (subset_closure ht) | |
| obtain ⟨J,Dnear,H,hDn,hH,hnf⟩ := actual_residual_full_child_near_farND | |
| hΨ hs hΨpos hsig hsig₁ W B total hW hJmask Cq hCq ((n+2:ℝ)*sig) T D ell hK | |
| obtain ⟨N₀,hnf⟩ := eventually_atTop.mp hnf | |
| refine ⟨2*(J+1)+n+3,Dnear,H,hDn,hH,N₀,?_⟩ | |
| intro e hN he hJe B₀ s hd hq ht₀ ha₀ hv₀ h hah hhN hdepth hX hY hoccX hoccY | |
| L₁ L₂ f g Q P Mn Mf hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP hn hm hfar | |
| let := hd.probability | |
| have hXlen := (mem_dyadicPaths e.a e.X).mp hX | |
| have hYlen := (mem_dyadicPaths e.a e.Y).mp hY | |
| have hXgrid : treeCell e.X=Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.X j:ℤ)) := by | |
| unfold treeCell; rw [hXlen]; rfl | |
| have hYgrid : treeCell e.Y=Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.Y j:ℤ)) := by | |
| unfold treeCell; rw [hYlen]; rfl | |
| have hx : e.ρ (Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.X j:ℤ)))≠0 := by | |
| rw [←hXgrid]; exact ne_of_gt hoccX | |
| have hy : e.ρ (Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.Y j:ℤ)))≠0 := by | |
| rw [←hYgrid]; exact ne_of_gt hoccY | |
| have hup (Z : CellPath (n+1)) (hZ : Z.length=e.a) (ho : 0<e.ρ (treeCell Z)) : | |
| (e.ρ (Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex Z j:ℤ)))).toReal≤ | |
| ((2:ℝ)^e.N)^(-e.mass e.a+(n+2:ℝ)*sig) := by | |
| have hz := (hd.cells e.a (by omega) Z hZ ho).2 | |
| rw [←Real.rpow_add (by positivity : (0:ℝ)<2^e.N)] at hz | |
| have hc : treeCell Z=Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex Z j:ℤ)) := by | |
| unfold treeCell; rw [hZ]; rfl | |
| rw [hc] at hz | |
| convert hz using 1; congr 1; push_cast; ring | |
| have hlow (Z : CellPath (n+1)) (hZ : Z.length=e.a) : | |
| ∀ b∈Finset.Icc (e.a+1) (h+ell+2),∀ z∈occupiedDescendantsND e.ρ e.a b (fun j => (pathIndex Z j:ℤ)), | |
| ((2:ℝ)^e.N)^(-e.mass b-(n+2:ℝ)*sig)≤(e.ρ (Attempt4FullFar.ambientGridCell b z)).toReal := by | |
| intro b hb z hz | |
| have hb' := Finset.mem_Icc.mp hb | |
| exact hd.occupiedDescendant_lower Z hZ (by omega) (by omega) z hz | |
| have hvN : (e.v:ℝ)-(e.a:ℝ)≤e.N := by | |
| have := (Nat.cast_le (α:=ℝ)).mpr (hv₀.trans ht₀) | |
| have := Nat.cast_nonneg (α:=ℝ) e.a | |
| linarith | |
| have hh := hnf e.N hN e.ρ e.ρ e.profile e.q e.a₀ e.t₀ e.a h e.v hq ht₀ ha₀ hv₀ | |
| L₁ L₂ e.mass e.C e.i f g Q P (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) | |
| e.v (e.mass e.a) (fun b => e.mass b) hf hg hfn hgn hL₁ hL₂ ht₁ ht₂ hQ hP hx hy hah hvN hdepth | |
| (hlow e.X hXlen) (hlow e.Y hYlen) (hup e.X hXlen hoccX) (hup e.Y hYlen hoccY) | |
| dsimp only at hh | |
| rw [←he,←hJe] at hh | |
| have hnear := recursive_near_max_le e h hah.le hXlen hYlen (ne_of_gt hoccX) (ne_of_gt hoccY) | |
| L₁ L₂ f g Q P Ψ Mn hn | |
| have hfars (b : ℕ) (hb : b∈Finset.Icc (e.a+1) (h+ell+2)) := | |
| recursive_far_max_le e b (by have := (Finset.mem_Icc.mp hb).1; omega) hXlen hYlen hK (hm b hb) (hfar b hb) | |
| have hdprof (b : ℕ) : e.mass b-(n+1:ℝ)*(b:ℝ)/(2*e.N)=e.profile b := by | |
| dsimp [RecursiveAngularND.profile,RecursiveEnvironmentND.profile,adjustedProfile] | |
| push_cast; ring | |
| have hn' := hnear ((original_occupied_nonemptyND e.ρ hah.le _ hx).product | |
| (original_occupied_nonemptyND e.ρ hah.le _ hy)) | |
| have hdom := add_le_add | |
| (add_le_add | |
| (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hn' (by positivity : 0≤Dnear^(4*total))) | |
| (by positivity : 0≤(20*(n+2:ℝ)+5)^(2*(n+1))* | |
| ((2:ℝ)^e.N)^(-2*((e.mass h-(n+1:ℝ)*(h:ℝ)/(2*e.N))-(e.mass e.a-(n+1:ℝ)*(e.a:ℝ)/(2*e.N)))+4*((n+2:ℝ)*e.sig)))) | |
| (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun b hb => | |
| mul_le_mul_of_nonneg_left (hfars b hb) (by positivity : 0≤ | |
| ((2:ℝ)^e.N)^(-2*((e.mass b-(n+1:ℝ)*(b:ℝ)/(2*e.N))-(e.mass e.a-(n+1:ℝ)*(e.a:ℝ)/(2*e.N)))+4*((n+2:ℝ)*e.sig))))) hH)) | |
| (le_refl (((2:ℝ)^e.N)^(-T))) | |
| have hfinal := hh.trans hdom | |
| conv_lhs at hfinal => unfold originalCellTransformND | |
| change treeCell e.X=GraphLiteralSupport.CurrentFalconer.ambientGridCell e.a (fun j => (pathIndex e.X j:ℤ)) at hXgrid | |
| change treeCell e.Y=GraphLiteralSupport.CurrentFalconer.ambientGridCell e.a (fun j => (pathIndex e.Y j:ℤ)) at hYgrid | |
| rw [←hXgrid,←hYgrid] at hfinal | |
| change e.energy L₁ L₂ f g Q P Ψ≤_ at hfinal | |
| simp only [hdprof] at hfinal | |
| rw [he] at hfinal | |
| convert hfinal using 1; ring_nf | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.DimensionalPolicy | |
| open PlanarFalconer | |
| theorem weighted_branch_energy_induction (A : ℕ → ℝ) (N q offset D : ℕ) | |
| (sig K lam eps C terminal floor error L E : ℝ) (root : BranchState) | |
| (energies : BranchState → ℝ → Prop) | |
| (hN : 0 < N) (hq : 0 < q) (hsig : 0 < sig) | |
| (hC : 1 ≤ C) (hL : 0 ≤ L) (hE : 0 ≤ E) | |
| (hvalid : ∀ s u l, s.Valid N sig → BranchStep A N q offset sig s l u → u.Valid N sig) | |
| (hdepth : ∀ n, ∀ b : PolicyBranch A N q offset sig n, b.state 0 = root → n ≤ D) | |
| (hterminal : ∀ s, s.Valid N sig → s.finish ≤ s.start+100*q → | |
| ∀ x, energies s x → x ≤ terminal) | |
| (hreduction : ∀ s, s.Valid N sig → | |
| (∃ n, ∃ b : PolicyBranch A N q offset sig n, b.state 0 = root ∧ b.state n = s) → | |
| s.start+100*q < s.finish → | |
| ∀ M : ℝ, 0 ≤ M → floor ≤ M → | |
| (∀ u l, BranchStep A N q offset sig s l u → ∀ x, energies u x → | |
| ((2:ℝ)^N)^(branchExponentND A K lam sig eps s l u)*x ≤ M) → | |
| ∀ x, energies s x → x ≤ C*M+error) | |
| (htermWeight : ∀ n, ∀ b : PolicyBranch A N q offset sig n, b.state 0 = root → | |
| b.weight K lam eps*terminal ≤ L) | |
| (hfloorWeight : ∀ n, ∀ b : PolicyBranch A N q offset sig n, b.state 0 = root → | |
| b.weight K lam eps*floor ≤ L) | |
| (herrorWeight : ∀ n, ∀ b : PolicyBranch A N q offset sig n, b.state 0 = root → | |
| b.weight K lam eps*error ≤ E) : | |
| ∀ n, ∀ b : PolicyBranch A N q offset sig n, b.state 0 = root → | |
| ∀ x, energies (b.state n) x → | |
| b.weight K lam eps*x ≤ (C+1)^(D-n+1)*(L+E) := by | |
| have haux : ∀ s : BranchState, ∀ n, ∀ b : PolicyBranch A N q offset sig n, | |
| b.state 0 = root → b.state n = s → ∀ x, energies s x → | |
| b.weight K lam eps*x ≤ (C+1)^(D-n+1)*(L+E) := by | |
| intro s | |
| induction s using (measure (fun s : BranchState => s.finish-s.start)).wf.induction with | |
| | h s ih => | |
| intro n b hroot hlast x hx | |
| have hs : s.Valid N sig := hlast ▸ b.valid n le_rfl | |
| have hW := b.weight_pos K lam eps | |
| have hsum : 0 ≤ L+E := add_nonneg hL hE | |
| have hpow : 1 ≤ (C+1)^(D-n) := one_le_pow₀ (by linarith) | |
| let H := (C+1)^(D-n)*(L+E) | |
| have hLH : L ≤ H := by dsimp only [H]; nlinarith | |
| have hEH : E ≤ H := by dsimp only [H]; nlinarith | |
| have hH : 0 ≤ H := hL.trans hLH | |
| have hHlast : H ≤ (C+1)^(D-n+1)*(L+E) := by | |
| rw [pow_succ] | |
| change H ≤ (C+1)^(D-n)*(C+1)*(L+E) | |
| nlinarith | |
| by_cases hw : s.finish ≤ s.start+100*q | |
| · have ht := hterminal s hs hw x hx | |
| exact ((mul_le_mul_of_nonneg_left ht hW.le).trans (htermWeight n b hroot)).trans | |
| (hLH.trans hHlast) | |
| · have hfloor : floor ≤ H/b.weight K lam eps := by | |
| apply (le_div_iff₀ hW).mpr | |
| simpa only [mul_comm] using (hfloorWeight n b hroot).trans hLH | |
| have hchildren : ∀ u l, BranchStep A N q offset sig s l u → ∀ y, energies u y → | |
| ((2:ℝ)^N)^(branchExponentND A K lam sig eps s l u)*y ≤ H/b.weight K lam eps := by | |
| intro u l hstep y hy | |
| have hu := hvalid s u l hs hstep | |
| have hstep' : BranchStep A N q offset sig (b.state n) l u := hlast ▸ hstep | |
| let b' := b.extend hu hstep' | |
| have hroot' : b'.state 0 = root := by | |
| change (b.extend hu hstep').state 0 = root | |
| rw [b.extend_state_le hu hstep' (Nat.zero_le n),hroot] | |
| have hlast' : b'.state (n+1) = u := b.extend_last_state hu hstep' | |
| have hnd := hdepth (n+1) b' hroot' | |
| have hi := ih u (hstep.width_decreases hN hq hsig hs hu) (n+1) b' hroot' hlast' y hy | |
| have harith : D-(n+1)+1 = D-n := by omega | |
| rw [harith] at hi | |
| change b'.weight K lam eps*y ≤ H at hi | |
| dsimp only [b'] at hi | |
| rw [b.weight_extend,hlast] at hi | |
| apply (le_div_iff₀ hW).mpr | |
| nlinarith only [hi] | |
| have hred := hreduction s hs ⟨n,b,hroot,hlast⟩ (by omega) (H/b.weight K lam eps) | |
| (div_nonneg hH hW.le) hfloor hchildren x hx | |
| have hh := mul_le_mul_of_nonneg_left hred hW.le | |
| have herr := herrorWeight n b hroot | |
| have hid : b.weight K lam eps*(C*(H/b.weight K lam eps)+error) = | |
| C*H+b.weight K lam eps*error := by field_simp | |
| rw [hid] at hh | |
| calc | |
| b.weight K lam eps*x ≤ C*H+E := by linarith | |
| _ ≤ (C+1)*H := by linarith | |
| _ = (C+1)^(D-n+1)*(L+E) := by dsimp only [H]; rw [pow_succ]; ring | |
| intro n b hr x hx | |
| exact haux (b.state n) n b hr rfl x hx | |
| end GraphLiteralSupport.CurrentFalconer.DimensionalPolicy | |
| end | |
| end | |
| end OAI | |