| 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 |
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| noncomputable section |
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| universe u |
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| namespace NoiselessRegression |
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| 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) |
|
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| 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 |
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| end NoiselessRegression |
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| end |
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| end OAI |
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|
