File size: 2,704 Bytes
b317718 | 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 | import OAI.Probability.GaussianRegression.Stream
namespace OAI
/-!
Finite-state sequential execution and terminal-state witnesses.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
lemma fin_prefix_snoc {X : Type*} {n T : ℕ} (h : n+1 ≤ T) (x : Fin T → X) :
(fun i : Fin (n+1) => x (Fin.castLE h i)) =
(Fin.snoc (fun i : Fin n => x (Fin.castLE (by omega) i))
(x ⟨n, by omega⟩) : Fin (n+1) → X) := by
ext i
refine Fin.lastCases ?_ (fun j => ?_) i
· simp only [Fin.snoc_last]
rfl
· simp
namespace Learner
variable {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
def forcedRun (L : Learner d M T Ω) (s : UnitSphere d) (ω : Ω) (x : Samples d T) : ℕ → State M
| 0 => (L.initialChoice ω).2
| n+1 =>
if hn : n < T then
(L.transition (n+1) ω (L.forcedRun s ω x n) (x ⟨n,hn⟩, ⟪x ⟨n,hn⟩,s.val⟫)).2
else L.forcedRun s ω x n
lemma forcedRun_eq_route (s : UnitSphere d) (ω : Ω) (x : Samples d T)
{n : ℕ} (hn : n ≤ T) :
L.forcedRun s ω x n =
streamRoute (fun t j obs => (L.transition t ω j obs).2) 0 n (L.initialChoice ω).2
((fun i => x (Fin.castLE hn i)), (fun i => ⟪s.val,x (Fin.castLE hn i)⟫)) := by
induction n with
| zero => rfl
| succ n ih =>
rw [forcedRun, dite_eq_left (by omega), ih (by omega)]
rw [fin_prefix_snoc hn x, fin_prefix_snoc hn (fun i => ⟪s.val,x i⟫), streamRoute_snoc]
simp only [zero_add, real_inner_comm]
lemma run_forced_witness (s : UnitSphere d) (ω : Ω) (x : Samples d T)
(n : ℕ) (hn : n ≤ T) :
(L.run s ω x n = .inl (L.forcedRun s ω x n)) ∨
∃ t ≤ n, L.run s ω x n = .inr (t, L.forcedRun s ω x t) := by
induction n with
| zero =>
simp only [run, start, forcedRun]
split
· exact Or.inr ⟨0, le_rfl, rfl⟩
· exact Or.inl rfl
| succ n ih =>
rcases ih (by omega) with h | ⟨t,ht,h⟩
· have hnt : n < T := by omega
rw [run, h]
simp only [dite_eq_left hnt, forcedRun]
split
· exact Or.inr ⟨n+1, le_rfl, by simp [forcedRun, hnt]⟩
· exact Or.inl rfl
· right
exact ⟨t, by omega, by rw [run, h]⟩
lemma terminal_forced_witness (s : UnitSphere d) (ω : Ω) (x : Samples d T) :
∃ t ≤ T, L.terminal s ω x = (t, L.forcedRun s ω x t) := by
rcases L.run_forced_witness s ω x T le_rfl with h | ⟨t,ht,h⟩
· exact ⟨T, le_rfl, by simp [terminal, h]⟩
· exact ⟨t, ht, by simp [terminal, h]⟩
end Learner
end NoiselessRegression
end
end OAI
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