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a2ee19a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CubeContinuation
import OAI.Probability.MemoryPrecision.StochasticContinuation
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
lemma stoch_sphere_mass {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N)
{δ : ℕ → Fin N → (Space d × ℝ) → Fin N → ℝ≥0∞}
{w : ℕ → Fin N → Sphere d → ℝ≥0∞}
(hδ : ∀ n j a, Measurable (fun obs => δ n j obs a))
(hδ1 : ∀ n j obs, ∑ a, δ n j obs a ≤ 1) (hw : ∀ n j, Measurable (w n j))
(hw1 : ∀ n j s, w n j s ≤ 1) {H α : ℝ} (hH : 0 < H)
(hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2)
(hlocal : ∀ n j, SphereBallBound (w n j) H α)
(b k : ℕ) (hk : k ≤ b*(d/8)) (n : ℕ) (j : Fin N) :
SphereBallBound (stochCont δ w Subtype.val n k j)
(H * RealStep d N ^ b) α := by
have hp : 0 < d/8 := by omega
have hK := realStep_one d hN
have hK0 := realStep_pos d hN
induction b generalizing k n j with
| zero =>
have : k = 0 := by omega
subst k
simpa [stochCont] using hlocal n j
| succ b ih =>
by_cases hk0 : k = 0
· subst k
apply (hlocal n j).mono
simpa [stochCont] using le_mul_of_one_le_right hH.le (one_le_pow₀ hK)
· let m := min k (d/8)
have hm : 1 ≤ m := by dsimp [m]; omega
have hmp : m ≤ d/8 := min_le_right _ _
have hmk : m ≤ k := min_le_left _ _
have hk' : k-m ≤ b*(d/8) := by
dsimp [m]
rw [Nat.succ_mul] at hk
omega
have hb := (Classical.choose_spec coefficient_step).2 d m N hd hm hmp hN α
(H * RealStep d N ^ b) hαlo hαhi (mul_pos hH (pow_pos hK0 b))
(fun a => stochCont δ w Subtype.val (n+m) (k-m) a)
(fun a => measurable_stochCont hδ hw measurable_subtype_coe _ _ a)
(fun a s => stochCont_le_one Subtype.val hδ1 hw1 _ _ a s)
(fun a => ih (k-m) hk' (n+m) a)
(fun a x => stochRoute δ n m j x a)
(fun a => measurable_stochRoute hδ n m j a)
(fun x => stochRoute_sum_le_one hδ1 n m j x)
have he : stochCont δ w Subtype.val n k j =
SphereBlock (fun a => stochCont δ w Subtype.val (n+m) (k-m) a)
(fun a x => stochRoute δ n m j x a) := by
funext s
simpa [Nat.add_sub_of_le hmk, SphereBlock, WeightedBlock] using
stochCont_unroll Subtype.val hδ n m (k-m) j s
rw [he]
convert hb using 1
change H * RealStep d N ^ (b+1) = RealStep d N * (H * RealStep d N ^ b)
rw [pow_succ]; ring
lemma stoch_cube_mass {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N)
{δ : ℕ → Fin N → (Space d × ℝ) → Fin N → ℝ≥0∞}
{w : ℕ → Fin N → Cube d → ℝ≥0∞}
(hδ : ∀ n j a, Measurable (fun obs => δ n j obs a))
(hδ1 : ∀ n j obs, ∑ a, δ n j obs a ≤ 1) (hw : ∀ n j, Measurable (w n j))
(hw1 : ∀ n j s, w n j s ≤ 1) {H : ℝ} (hH : 0 < H)
(hlocal : ∀ n j, CubeLocal (w n j) H)
(b k : ℕ) (hk : k ≤ b*((d/2)/4)) (n : ℕ) (j : Fin N) :
CubeLocal (stochCont δ w (CubeEmbedding d) n k j)
(H * CubeStep d N ^ b) := by
have hp : 0 < (d/2)/4 := by omega
have hK := cubeStep_one d hN
have hK0 := cubeStep_pos d hN
induction b generalizing k n j with
| zero =>
have : k = 0 := by omega
subst k
simpa [stochCont] using hlocal n j
| succ b ih =>
by_cases hk0 : k = 0
· subst k
apply (hlocal n j).mono
simpa [stochCont] using le_mul_of_one_le_right hH.le (one_le_pow₀ hK)
· let m := min k ((d/2)/4)
have hm : 1 ≤ m := by dsimp [m]; omega
have hmp : m ≤ (d/2)/4 := min_le_right _ _
have hmk : m ≤ k := min_le_left _ _
have hk' : k-m ≤ b*((d/2)/4) := by
dsimp [m]
rw [Nat.succ_mul] at hk
omega
have hb := (Classical.choose_spec cube_block).2 d m N hd hm hmp hN
(H * CubeStep d N ^ b) (mul_pos hH (pow_pos hK0 b)).le
(fun a => stochCont δ w (CubeEmbedding d) (n+m) (k-m) a)
(fun a => measurable_stochCont hδ hw (cubeEmbedding_measurable d) _ _ a)
(fun a s => stochCont_le_one (CubeEmbedding d) hδ1 hw1 _ _ a s)
(fun a => ih (k-m) hk' (n+m) a)
(fun a x => stochRoute δ n m j x a)
(fun a => measurable_stochRoute hδ n m j a)
(fun x => stochRoute_sum_le_one hδ1 n m j x)
have he : stochCont δ w (CubeEmbedding d) n k j =
WeightedBlock (CubeEmbedding d)
(fun a => stochCont δ w (CubeEmbedding d) (n+m) (k-m) a)
(fun a x => stochRoute δ n m j x a) := by
funext s
simpa [Nat.add_sub_of_le hmk, WeightedBlock] using
stochCont_unroll (CubeEmbedding d) hδ n m (k-m) j s
rw [he]
convert hb using 1
change H * CubeStep d N ^ (b+1) = CubeStep d N * (H * CubeStep d N ^ b)
rw [pow_succ]; ring
end MemoryPrecision
end
end OAI
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