openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Recursion /ActualDistanceRecursionBound.lean
| import OAI.MeasureTheory.Falconer.Angular.ActualAngularRecursionBound | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set MeasureTheory | |
| open scoped ContDiff | |
| theorem PositiveDistanceData.mul_prefixLowMaximum_le (d : PositiveDistanceData) | |
| {c M : ℝ} (hc : 0 < c) (hM : 0 ≤ M) | |
| (hb : ∀ v ∈ d.prefixLowDepths, | |
| c*({d with t := v,i := d.i+1} : PositiveDistanceData).coincidence.toReal ≤ M) : | |
| c*d.prefixLowMaximum ≤ M := by | |
| have hh := d.prefixLowMaximum_le (div_nonneg hM hc.le) | |
| (fun v hv => (le_div_iff₀ hc).mpr (by simpa only [mul_comm] using hb v hv)) | |
| exact (mul_le_mul_of_nonneg_left hh hc.le).trans_eq (by field_simp) | |
| theorem uniform_actual_distance_recursion_bound {lam sig kap eps T : ℝ} | |
| (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) | |
| (hgap : sig < kap/1000) (heps : 0 < eps) (hT : 0 < T) | |
| {Ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hpos : ∀ x, 0 ≤ Ψ x) | |
| (hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) | |
| (hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) : | |
| ∃ Kinv N₀ : ℕ, ∀ K ≥ Kinv, ∀ d : PositiveDistanceData, | |
| d.lam = lam → d.sig = sig → d.q = sourcePolicyBlock d.N kap → | |
| d.J = sourceSmoothingLevels sig kap → N₀ ≤ d.N → | |
| d.Q.side = 1 → d.ρ d.Q.carrierᶜ = 0 → d.RecursiveCells → | |
| d.branchState.Valid d.N d.sig → d.a+100*d.q < d.t → d.i+1 ≤ d.J → | |
| ∀ B : ℕ, ∀ δ : ℝ, RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass → | |
| ∀ M : ℝ, 0 ≤ M → ((2:ℝ)^d.N)^(12*lam+eps) ≤ M → | |
| (∀ t i l, BranchStep d.branchProfile d.N d.q d.sig d.branchState l (.D d.a t i) → | |
| ((2:ℝ)^d.N)^(branchExponent d.branchProfile lam sig eps d.branchState l (.D d.a t i)) * | |
| ({d with t := t,i := i} : PositiveDistanceData).coincidence.toReal ≤ M) → | |
| (∀ p v, ∀ row ∈ sourceProductVertices d.Q d.Q d.ρ d.ρ d.X d.Y (p-d.a), | |
| BranchStep d.branchProfile d.N d.q d.sig d.branchState .high (.H d.a d.t p v d.i) → | |
| ∀ k : Fin 2 → PolicyDescriptor → ℕ, (∀ l, ∑ e ∈ d.prefixRemainingList p, k l e ≤ K) → | |
| ∀ h : Fin 2 → Plane → ℂ, (∀ l, Measurable (h l)) → (∀ l x, ‖h l x‖ ≤ 1) → | |
| ((2:ℝ)^d.N)^(branchExponent d.branchProfile lam sig eps d.branchState .high (.H d.a d.t p v d.i)) * | |
| ((d.angularSuccessor p v row).energy k h Ψ).toReal ≤ M) → | |
| d.coincidence ≠ ⊤ ∧ d.coincidence.toReal ≤ 3*M+((2:ℝ)^d.N)^(eps-T) := by | |
| obtain ⟨N₁,h₁⟩ := source_positive_distance_suffix hlam hsig hkap heps | |
| obtain ⟨Kinv,N₂,h₂⟩ := source_positive_distance_prefix hlam hsig hkap hgap heps hT hΨ hpos hs hone | |
| refine ⟨Kinv,max N₁ N₂,?_⟩ | |
| intro K hK d hdlam hdsig hdq hdJ hN hQ hcarry hcells hv hw hi B δ hprof M hM hfloor hD hH | |
| let := hprof.probability | |
| have hfinite := d.coincidence_ne_top | |
| change d.a ≤ d.t ∧ d.t ≤ d.N at hv | |
| let p := policyDepth d.branchProfile d.N d.q d.a d.t | |
| have hRp : 0 < (2:ℝ)^d.N := by positivity | |
| by_cases hp : d.a+d.t < 2*p | |
| · have hbound := h₁ d hdlam hdsig hdq hdJ (by omega) hQ | |
| (hcells.stronglySeparated hQ) hcells.1 hcells.2.1 B δ hprof hw hv.2 hi hp | |
| have hstep : BranchStep d.branchProfile d.N d.q d.sig d.branchState .suffix (.D d.a p (d.i+1)) := | |
| BranchStep.suffix hw hp | |
| have hm := hD p (d.i+1) .suffix hstep | |
| have hboundR := ENNReal.toReal_mono | |
| (ENNReal.mul_ne_top ENNReal.ofReal_ne_top | |
| ({d with t := p,i := d.i+1} : PositiveDistanceData).coincidence_ne_top) hbound | |
| rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity)] at hboundR | |
| simp only [branchExponent,branchCost,PositiveDistanceData.branchState,BranchState.start,BranchState.finish, | |
| sub_self,mul_zero,zero_add,reduceCtorEq,ite_false] at hm | |
| have hboundM : d.coincidence.toReal ≤ M := hboundR.trans hm | |
| refine ⟨hfinite,hboundM.trans ?_⟩ | |
| linarith [Real.rpow_pos_of_pos hRp (eps-T)] | |
| · have hp' : 2*p ≤ d.a+d.t := by omega | |
| have hbound := h₂ d hdlam hdsig hdq hdJ (by omega) hQ hcarry hcells.1 hcells.2.1 | |
| hcells.2.2.1 hcells.2.2.2 B δ hprof hw hv.2 hi hp' | |
| have hlow : ((2:ℝ)^d.N)^eps*d.prefixLowMaximum ≤ M := by | |
| apply d.mul_prefixLowMaximum_le (by positivity) hM | |
| intro v hv | |
| have hm := hD v (d.i+1) .low (d.low_branch_transition hw hv) | |
| simpa only [branchExponent,branchCost,PositiveDistanceData.branchState,BranchState.start,sub_self,mul_zero, | |
| neg_zero,zero_add,ite_true] using hm | |
| let E := -2*(d.branchProfile p-d.branchProfile d.a)+height d.branchProfile d.a p+4000*(lam+sig) | |
| have hhigh : ((2:ℝ)^d.N)^(E+eps)*d.prefixSuccessorSupremum Kinv p Ψ ≤ M := by | |
| have hsup := d.prefixSuccessorSupremum_le Kinv p Ψ | |
| (div_nonneg hM (by positivity : 0 ≤ ((2:ℝ)^d.N)^(E+eps))) (by | |
| intro v hv hg row hr k hk h hh hhb | |
| have hm := hH p v row hr (d.high_branch_transition hw hp' hv hg row) k | |
| (fun l => (hk l).trans hK) h hh hhb | |
| apply (le_div_iff₀ (by positivity : 0 < ((2:ℝ)^d.N)^(E+eps))).mpr | |
| simpa only [branchExponent,branchCost,PositiveDistanceData.branchState,BranchState.start, | |
| reduceCtorEq,ite_false,E,mul_comm] using hm) | |
| exact (mul_le_mul_of_nonneg_left hsup (by positivity)).trans_eq (by field_simp) | |
| have hfloor' : ((2:ℝ)^d.N)^eps*((2:ℝ)^d.N)^(12*lam) ≤ M := by | |
| rw [← Real.rpow_add hRp,add_comm eps] | |
| exact hfloor | |
| have hhigh' : ((2:ℝ)^d.N)^eps*(((2:ℝ)^d.N)^E*d.prefixSuccessorSupremum Kinv p Ψ) ≤ M := by | |
| rw [← mul_assoc,← Real.rpow_add hRp,add_comm eps] | |
| exact hhigh | |
| have herr : ((2:ℝ)^d.N)^eps*((2:ℝ)^d.N)^(-T) = ((2:ℝ)^d.N)^(eps-T) := by | |
| rw [← Real.rpow_add hRp,sub_eq_add_neg] | |
| refine ⟨hfinite,hbound.trans ?_⟩ | |
| change ((2:ℝ)^d.N)^eps * (((2:ℝ)^d.N)^(12*lam)+d.prefixLowMaximum+ | |
| ((2:ℝ)^d.N)^E*d.prefixSuccessorSupremum Kinv p Ψ+((2:ℝ)^d.N)^(-T)) ≤ _ | |
| simp only [mul_add] | |
| rw [herr] | |
| linarith | |
| end PlanarFalconer | |
| end OAI | |