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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | 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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