| import OAI.MeasureTheory.Falconer.Recursion.BranchPotential |
| import OAI.MeasureTheory.Falconer.Recursion.SourceBranchDepth |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| theorem branch_nonlow_indicator_sum (label : ℕ → BranchLabel) (n : ℕ) : |
| (∑ j ∈ Finset.range n, if label j = .low then 0 else 1) = branchNonlowCount label n := by |
| have htot := branch_count_total label n |
| have hsum : branchCount label n .low + |
| (∑ j ∈ Finset.range n, if label j = .low then 0 else 1) = n := by |
| simp only [branchCount,← Finset.sum_add_distrib] |
| calc |
| _ = ∑ _j ∈ Finset.range n, (1:ℕ) := by |
| apply Finset.sum_congr rfl |
| intro j _ |
| by_cases hj : label j = .low < |
| _ = n := by simp |
| omega |
|
|
| theorem branch_nonlow_loss_sum (label : ℕ → BranchLabel) (n : ℕ) (loss : ℝ) : |
| (∑ j ∈ Finset.range n, if label j = .low then 0 else loss) = |
| (branchNonlowCount label n:ℝ)*loss := by |
| have hcast := congrArg (fun k : ℕ => (k:ℝ)) (branch_nonlow_indicator_sum label n) |
| simp only [Nat.cast_sum,Nat.cast_ite,Nat.cast_zero,Nat.cast_one] at hcast |
| calc |
| _ = ∑ j ∈ Finset.range n, (if label j = .low then (0:ℝ) else 1)*loss := by |
| apply Finset.sum_congr rfl |
| intro j _ |
| by_cases hj : label j = .low < |
| _ = _ := by rw [← Finset.sum_mul,hcast] |
|
|
| noncomputable def branchExponent (A : ℕ → ℝ) (lam sig eps : ℝ) |
| (s : BranchState) (label : BranchLabel) (u : BranchState) : ℝ := |
| -2*(A u.start-A s.start)+branchCost A s label u+ |
| (if label = .low then 0 else 4000*(lam+sig))+eps |
|
|
| theorem branch_exponent_sum (A : ℕ → ℝ) (lam sig eps : ℝ) |
| (state : ℕ → BranchState) (label : ℕ → BranchLabel) (n : ℕ) : |
| (∑ j ∈ Finset.range n, branchExponent A lam sig eps (state j) (label j) (state (j+1))) = |
| 2*A (state 0).start-2*A (state n).start+ |
| (∑ j ∈ Finset.range n, branchCost A (state j) (label j) (state (j+1)))+ |
| (branchNonlowCount label n:ℝ)*(4000*(lam+sig))+(n:ℝ)*eps := by |
| simp only [branchExponent,Finset.sum_add_distrib] |
| rw [profile_increments_telescope A (fun j => (state j).start) n,branch_nonlow_loss_sum] |
| simp |
|
|
| theorem source_branch_exponent {A : ℕ → ℝ} {N n : ℕ} {lam sig kap eta : ℝ} |
| (hN : 0 < N) (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) (hkap1 : kap ≤ 1) |
| (heta : 0 < eta) (hceil : ⌈10*sig*(N:ℝ)⌉₊ < sourcePolicyBlock N kap) |
| (hslack : sig*(N:ℝ) ≤ sourcePolicyBlock N kap) |
| (hqratio : (sourcePolicyBlock N kap:ℝ)/(N:ℝ) ≤ 2*kap) |
| (hA : DiscreteProfileLipschitz A N) |
| (b : PolicyBranch A N (sourcePolicyBlock N kap) sig n) |
| (hinit : (b.state 0).start+10*sourcePolicyBlock N kap ≤ (b.state 0).finish) : |
| let eps := eta/(2*(1+20/kap+1/sig)) |
| (∑ j ∈ Finset.range n, branchExponent A lam sig eps (b.state j) (b.label j) (b.state (j+1))) + |
| (200*(sourcePolicyBlock N kap:ℝ)/N+12*lam+eta/2) ≤ |
| 2*A (b.state 0).start-potential A (b.state 0).start (b.state 0).finish |
| (10*sourcePolicyBlock N kap)+500*kap+10^6*(lam+sig)/kap+eta := by |
| dsimp only |
| have hq := sourcePolicyBlock_pos hN hkap |
| have hc := b.costs_le_final_start hN hq hslack hA hinit |
| have hb := source_branch_count_bounds hN hsig hkap hkap1 hceil b |
| let D : ℝ := 1+20/kap+1/sig |
| have hD : 0 < D := by dsimp [D] |
| have heps : 0 ≤ eta/(2*D) := by positivity |
| have htime : (n:ℝ)*(eta/(2*D)) ≤ eta/2 := by |
| have hn : (n:ℝ) ≤ D := by dsimp [D] |
| calc |
| _ ≤ D*(eta/(2*D)) := mul_le_mul_of_nonneg_right hn heps |
| _ = eta/2 := by field_simp |
| have hloss : (branchNonlowCount b.label n:ℝ)*(4000*(lam+sig)) ≤ 80000*(lam+sig)/kap := by |
| calc |
| _ ≤ (20/kap)*(4000*(lam+sig)) := mul_le_mul_of_nonneg_right hb.1 (by positivity) |
| _ = _ := by ring |
| have hlamK : lam ≤ (lam+sig)/kap := by |
| apply (le_div_iff₀ hkap).mpr |
| have hh := mul_le_mul_of_nonneg_left hkap1 hlam |
| linarith |
| have hbudget : 0 ≤ (lam+sig)/kap := by positivity |
| rw [branch_exponent_sum] |
| dsimp only [D] at htime |
| simp only [mul_div_assoc] at hc hloss ⊢ |
| nlinarith |
|
|
| theorem uniform_source_branch_exponent {lam sig kap eta : ℝ} |
| (hlam : 0 ≤ lam) (hsig : 0 < sig) (hkap : 0 < kap) (hkap1 : kap ≤ 1) |
| (hgap : sig < kap/1000) (heta : 0 < eta) : |
| ∃ N₀ : ℕ, 0 < N₀ ∧ ∀ N ≥ N₀, ∀ A : ℕ → ℝ, |
| DiscreteProfileLipschitz A N → ∀ n : ℕ, |
| ∀ b : PolicyBranch A N (sourcePolicyBlock N kap) sig n, |
| (b.state 0).start+10*sourcePolicyBlock N kap ≤ (b.state 0).finish → |
| (∑ j ∈ Finset.range n, branchExponent A lam sig (eta/(2*(1+20/kap+1/sig))) |
| (b.state j) (b.label j) (b.state (j+1))) + |
| (200*(sourcePolicyBlock N kap:ℝ)/N+12*lam+eta/2) ≤ |
| 2*A (b.state 0).start-potential A (b.state 0).start (b.state 0).finish |
| (10*sourcePolicyBlock N kap)+500*kap+10^6*(lam+sig)/kap+eta := by |
| obtain ⟨N₁,hN₁,h₁⟩ := uniform_source_ceiling_slack hsig hgap |
| obtain ⟨N₂,hN₂⟩ := exists_nat_gt (1/kap) |
| refine ⟨max N₁ N₂,by omega,?_⟩ |
| intro N hN A hA n b hi |
| have hNp : 0 < N := by omega |
| have hNR : (0:ℝ) < N := Nat.cast_pos.mpr hNp |
| have hkn : 1 < kap*(N:ℝ) := by |
| have hh := (div_lt_iff₀ hkap).mp (hN₂.trans_le (Nat.cast_le.mpr (show N₂ ≤ N by omega))) |
| linarith |
| have hq : (sourcePolicyBlock N kap:ℝ) < kap*N+1 := |
| Nat.ceil_lt_add_one (mul_nonneg hkap.le hNR.le) |
| have hqr : (sourcePolicyBlock N kap:ℝ)/(N:ℝ) ≤ 2*kap := |
| (div_le_iff₀ hNR).mpr (by linarith) |
| exact source_branch_exponent hNp hlam hsig hkap hkap1 heta |
| (h₁ N (by omega)).1 (h₁ N (by omega)).2.2 hqr hA b hi |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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