| import OAI.MeasureTheory.Falconer.Recursion.BranchExponent |
| import OAI.MeasureTheory.Falconer.Angular.AngularUniformAbsorption |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| theorem DiscreteProfileLipschitz.increment_le_one {A : ℕ → ℝ} {N i j : ℕ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) (hi : i ≤ N) (hj : j ≤ N) : |
| |A i-A j| ≤ 1 := by |
| apply (hA i hi j hj).trans |
| apply (div_le_one (Nat.cast_pos.mpr hN)).mpr |
| apply abs_le.mpr |
| have hi' : (i:ℝ) ≤ N := Nat.cast_le.mpr hi |
| have hj' : (j:ℝ) ≤ N := Nat.cast_le.mpr hj |
| constructor <;> linarith [Nat.cast_nonneg (α := ℝ) i,Nat.cast_nonneg (α := ℝ) j] |
|
|
| theorem DiscreteProfileLipschitz.abs_le_one {A : ℕ → ℝ} {N i : ℕ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) (hzero : A 0 = 0) (hi : i ≤ N) : |
| |A i| ≤ 1 := by |
| simpa only [hzero,sub_zero] using hA.increment_le_one hN hi (Nat.zero_le N) |
|
|
| theorem DiscreteProfileLipschitz.minimum_cost_le_one {A : ℕ → ℝ} {N a t i : ℕ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) |
| (hat : a ≤ t) (ht : t ≤ N) (hi : i ≤ N) : |
| A i-intervalMin A a t ≤ 1 := by |
| obtain ⟨j,_,hjt,hmin⟩ := intervalMin_attained A hat |
| rw [hmin] |
| exact (le_abs_self _).trans (hA.increment_le_one hN hi (hjt.trans ht)) |
|
|
| theorem BranchStep.cost_le_one {A : ℕ → ℝ} {N q : ℕ} {sig : ℝ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) (hq : 0 < q) |
| {s u : BranchState} {l : BranchLabel} (h : BranchStep A N q sig s l u) |
| (hs : s.Valid N sig) (hu : u.Valid N sig) : branchCost A s l u ≤ 1 := by |
| have hn := h.nested hq hs |
| have hsd := BranchState.valid_depths hs |
| have hud := BranchState.valid_depths hu |
| cases h <;> simp only [branchCost] |
| · exact hA.minimum_cost_le_one hN hn.2 hsd.2 (hud.2.trans (le_refl _)) |
| · norm_num |
| · exact hA.minimum_cost_le_one hN hn.1 (hud.1.trans hud.2) (hud.1.trans hud.2) |
| · norm_num |
| · norm_num |
| · exact hA.minimum_cost_le_one hN hn.1 (hud.1.trans hud.2) (hud.1.trans hud.2) |
|
|
| theorem BranchStep.exponent_le_four {A : ℕ → ℝ} {N q : ℕ} {lam sig eps : ℝ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) (hq : 0 < q) |
| (hloss : 4000*(lam+sig) ≤ 1) |
| {s u : BranchState} {l : BranchLabel} (h : BranchStep A N q sig s l u) |
| (hs : s.Valid N sig) (hu : u.Valid N sig) : |
| branchExponent A lam sig eps s l u ≤ 4+eps := by |
| have hc := h.cost_le_one hA hN hq hs hu |
| have hsd := BranchState.valid_depths hs |
| have hud := BranchState.valid_depths hu |
| have hi := abs_le.mp (hA.increment_le_one hN (hud.1.trans hud.2) (hsd.1.trans hsd.2)) |
| unfold branchExponent |
| split_ifs <;> linarith |
|
|
| theorem PolicyBranch.ancestor_exponent_bound {A : ℕ → ℝ} {N q n : ℕ} {lam sig eps : ℝ} |
| (b : PolicyBranch A N q sig n) (hA : DiscreteProfileLipschitz A N) |
| (hN : 0 < N) (hq : 0 < q) (hloss : 4000*(lam+sig) ≤ 1) : |
| (∑ j ∈ Finset.range n, branchExponent A lam sig eps (b.state j) (b.label j) (b.state (j+1))) ≤ |
| (n:ℝ)*(4+eps) := by |
| calc |
| _ ≤ ∑ _j ∈ Finset.range n, (4+eps) := by |
| apply Finset.sum_le_sum |
| intro j hj |
| have hj := Finset.mem_range.mp hj |
| exact (b.step j hj).exponent_le_four hA hN hq hloss (b.valid j hj.le) |
| (b.valid (j+1) (by omega)) |
| _ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] |
|
|
| theorem source_error_exponent {sig kap T eps n : ℝ} |
| (hsig : 0 < sig) (hkap : 0 < kap) (hn : 0 ≤ n) |
| (hdepth : n ≤ 20/kap+1/sig) (heps : eps ≤ 1) |
| (hT : 1000*(1/sig+1/kap)+1000 < T) : -T+n*(4+eps) < -10 := by |
| have hstep : n*(4+eps) ≤ 5*n := by nlinarith |
| have hk : 0 < 1/kap := by positivity |
| have hs : 0 < 1/sig := by positivity |
| have he : 20/kap = 20*(1/kap) := by ring |
| rw [he] at hdepth |
| linarith |
|
|
| theorem uniform_source_error_branch {lam sig kap T eps : ℝ} |
| (hsig : 0 < sig) (hkap : 0 < kap) (hkap1 : kap ≤ 1) (hgap : sig < kap/1000) |
| (hloss : 4000*(lam+sig) ≤ 1) (heps : eps ≤ 1) |
| (hT : 1000*(1/sig+1/kap)+1000 < T) : |
| ∃ 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).index = 0 → |
| ((2:ℝ)^N)^(-T+(∑ j ∈ Finset.range n, |
| branchExponent A lam sig eps (b.state j) (b.label j) (b.state (j+1)))) ≤ |
| ((2:ℝ)^N)^(-10:ℝ) := by |
| obtain ⟨N₀,hN₀,h₀⟩ := uniform_source_branch_depth hsig hkap hkap1 hgap |
| refine ⟨N₀,hN₀,?_⟩ |
| intro N hN A hA n b hi |
| have hNp : 0 < N := hN₀.trans_le hN |
| have hc := b.ancestor_exponent_bound (eps := eps) hA hNp (sourcePolicyBlock_pos hNp hkap) hloss |
| have hd := (h₀ N hN A n b hi).2.2.1 |
| have he := source_error_exponent hsig hkap (Nat.cast_nonneg n) hd heps hT |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) |
| linarith |
|
|
| theorem source_profile_exponent_lower {A : ℕ → ℝ} {N a t Q : ℕ} |
| (hA : DiscreteProfileLipschitz A N) (hN : 0 < N) (hzero : A 0 = 0) |
| (hQ : 0 < Q) (hw : a+Q ≤ t) (ht : t ≤ N) : |
| -4 ≤ 2*A a-potential A a t Q := by |
| have ha := abs_le.mp (hA.abs_le_one hN hzero (show a ≤ N by omega)) |
| have hb := abs_le.mp (hA.abs_le_one hN hzero (show a+Q ≤ N by omega)) |
| have hp := potential_le A (Q := Q) (le_refl a) (by omega : a < a+Q) hw (le_refl _) |
| linarith |
|
|
| theorem uniform_recursion_count_subpower (D : ℕ) {eta : ℝ} (heta : 0 < eta) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, (4*(N:ℝ)+1)^D ≤ ((2:ℝ)^N)^eta := |
| fixed_polynomial_subpower D heta |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
