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f18e7de | 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 | import OAI.MathematicalPhysics.AlternatingFlow.EffectiveProfiles
import OAI.MathematicalPhysics.AlternatingFlow.Names
namespace OAI
section DecoderBoundsDevelopment
open scoped BigOperators Topology ContDiff
open Filter
namespace AlternatingNS.Quantitative
lemma linear_id_jet (r : ℕ) {x : ℝ} (hx : |x| ≤ 1) : |iteratedDeriv r id x| ≤ 1 := by
rw [iteratedDeriv_id]; split_ifs <;> simp_all
lemma transition_affine (r : ℕ) (a d : ℝ) :
iteratedDeriv r (fun x => Real.smoothTransition ((x-a)/d)) =
fun x => d⁻¹ ^ r * iteratedDeriv r Real.smoothTransition ((x-a)/d) := by
have h : (fun x => Real.smoothTransition ((x-a)/d)) =
(fun x => Real.smoothTransition (d⁻¹ * (x-a))) := by funext x; congr 1; ring
rw [h, iteratedDeriv_comp_sub_const r (fun z => Real.smoothTransition (d⁻¹ * z)) a,
iteratedDeriv_comp_const_mul
(Real.smoothTransition.contDiff.of_le (WithTop.coe_le_coe.mpr le_top))]
simp only [div_eq_mul_inv, mul_comm]
lemma periodizer_jet_bound (r : ℕ) (a c : ℝ) (ha : 0 < a) (hac : a < c) (hc : c < 1)
(x : ℝ) :
‖iteratedFDeriv ℝ r (Profiles.periodizer a c) x‖ ≤
1 + (c-a)⁻¹ ^ r * transitionBound r := by
rw [norm_iteratedFDeriv_eq_norm_iteratedDeriv, Real.norm_eq_abs]
have hnon : 0 ≤ (c-a)⁻¹ ^ r * (transitionBound r : ℝ) := by positivity
by_cases hx : x = (⌊x⌋ : ℝ)
· have he : Profiles.periodizer a c =ᶠ[𝓝 x] (fun y => y - (⌊x⌋ : ℝ)) := by
have hlo : (⌊x⌋ : ℝ) + (c - 1) < x := by linarith
have hhi : x < (⌊x⌋ : ℝ) + a := by linarith
filter_upwards [Ioo_mem_nhds hlo hhi] with y hy
have hy' : y - (⌊x⌋ : ℝ) ∈ Set.Ioo (c - 1) a := by
constructor <;> linarith [hy.1, hy.2]
calc
Profiles.periodizer a c y = Profiles.periodizer a c ((⌊x⌋ : ℝ) + (y - (⌊x⌋ : ℝ))) := by congr 1; ring
_ = Profiles.periodizer a c (y - (⌊x⌋ : ℝ)) := Profiles.periodizer_int_add a c _ _
_ = y - (⌊x⌋ : ℝ) := Profiles.periodizer_near_zero a c ha hac hc _ hy'
rw [he.iteratedDeriv_eq r]
change |iteratedDeriv r (fun y => id (y - (⌊x⌋ : ℝ))) x| ≤ _
rw [iteratedDeriv_comp_sub_const]
exact (linear_id_jet r (by rw [hx]; simp)).trans (le_add_of_nonneg_right hnon)
· have he : Profiles.periodizer a c =ᶠ[𝓝 x]
(fun y => (y - (⌊x⌋ : ℝ)) - Real.smoothTransition ((y - (⌊x⌋ : ℝ)-a)/(c-a))) := by
have hlo : (⌊x⌋ : ℝ) < x := (Int.floor_le x).lt_of_ne (Ne.symm hx)
filter_upwards [Ioo_mem_nhds hlo (Int.lt_floor_add_one x)] with y hy
have hy' : ⌊y⌋ = ⌊x⌋ := Int.floor_eq_iff.2 ⟨hy.1.le, hy.2⟩
simp only [Profiles.periodizer, Int.fract, hy']
rw [he.iteratedDeriv_eq r]
change |iteratedDeriv r (fun y => ((id : ℝ → ℝ) - fun z => Real.smoothTransition ((z-a)/(c-a)))
(y - (⌊x⌋ : ℝ))) x| ≤ _
rw [iteratedDeriv_comp_sub_const]
dsimp only
rw [iteratedDeriv_sub contDiffAt_id
(show ContDiffAt ℝ (r : ℕ∞ω) (fun z => Real.smoothTransition ((z-a)/(c-a))) (x - (⌊x⌋ : ℝ)) from by fun_prop),
transition_affine]
calc
_ ≤ |iteratedDeriv r id (x - (⌊x⌋ : ℝ))| +
(c-a)⁻¹ ^ r * |iteratedDeriv r Real.smoothTransition ((x - (⌊x⌋:ℝ)-a)/(c-a))| := by
simpa only [Real.norm_eq_abs, abs_mul, abs_of_nonneg (pow_nonneg (inv_nonneg.mpr (sub_pos.mpr hac).le) r)] using
norm_sub_le (iteratedDeriv r id (x - (⌊x⌋ : ℝ)))
((c-a)⁻¹ ^ r * iteratedDeriv r Real.smoothTransition ((x - (⌊x⌋:ℝ)-a)/(c-a)))
_ ≤ _ := add_le_add (linear_id_jet r (by
rw [abs_of_nonneg (sub_nonneg.mpr (Int.floor_le x))]
linarith [Int.lt_floor_add_one x]))
(mul_le_mul_of_nonneg_left
(by simpa only [norm_iteratedFDeriv_eq_norm_iteratedDeriv, Real.norm_eq_abs] using
transition_jet_bound r ((x - (⌊x⌋:ℝ)-a)/(c-a)))
(pow_nonneg (inv_nonneg.mpr (sub_pos.mpr hac).le) r))
def decoderBound (b r : ℕ) : ℕ := 1 + (3 * (b - 1)) ^ r * transitionBound r
lemma decoder_jet_bound (b : ℕ) (hb : 2 ≤ b) (r : ℕ) (x : ℝ) :
‖iteratedFDeriv ℝ r (Profiles.decoder b) x‖ ≤ decoderBound b r := by
have hm0 := Encoding.tailMax_nonneg b hb
have hm1 := Encoding.tailMax_lt_one b hb
have hgap : (((Encoding.tailMax b + 2) / 3) - ((2 * Encoding.tailMax b + 1) / 3))⁻¹ =
(3 * (b - 1) : ℕ) := by
have hbn : (b - 1 : ℕ) = (b : ℝ) - 1 := by exact_mod_cast Nat.cast_sub (by omega : 1 ≤ b)
rw [Nat.cast_mul, hbn]
have h : (b : ℝ) - 1 ≠ 0 := by
have hb' : (2 : ℝ) ≤ b := by exact_mod_cast hb
linarith
unfold Encoding.tailMax
field_simp
ring
have h := periodizer_jet_bound r ((2 * Encoding.tailMax b + 1) / 3)
((Encoding.tailMax b + 2) / 3) (by linarith) (by linarith) (by linarith) x
rw [hgap] at h
simpa only [Profiles.decoder, decoderBound, Nat.cast_add, Nat.cast_mul, Nat.cast_one, Nat.cast_pow] using h
end AlternatingNS.Quantitative
end DecoderBoundsDevelopment
end OAI
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