| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerKeyLocalityLemmas |
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| namespace OAI |
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| section |
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| namespace PerfectCompleteness.LowerCutScalar |
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| open RecursiveSpaces DescendantSpaces RecursiveSampler TerminalCalls |
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| noncomputable section |
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| universe u w |
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| variable {branch : Nat → Nat} {n m : Nat} |
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| abbrev Values (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) |
| (p : Path branch n (m + 1)) (A : Slots branch n → Type u) := |
| TerminalIndex repeats p → space 𝕜 branch (m + 1) (p.family A) |
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| def reconstruct (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : |
| {n m : Nat} → (p : Path branch n (m + 1)) → (A : Slots branch n → Type u) → |
| Values 𝕜 repeats p A → CutTerminalSplit.ExteriorTape 𝕜 repeats p A → |
| space 𝕜 branch n A |
| | _, _, .refl _, _, values, _ => values () |
| | n + 1, _, .step i p, A, values, exterior => |
| RecursiveSampler.combine 𝕜 A i (repeats (n + 1)) |
| (fun j => exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩) |
| (fun h => reconstruct 𝕜 repeats p (childFamily A i) |
| (fun terminal => values ((h, false), terminal)) |
| (CutSamplerReplayGrouping.subExterior 𝕜 repeats i p A (h, false) exterior)) |
| (fun h => reconstruct 𝕜 repeats p (childFamily A i) |
| (fun terminal => values ((h, true), terminal)) |
| (CutSamplerReplayGrouping.subExterior 𝕜 repeats i p A (h, true) exterior)) |
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| @[simp] theorem reconstruct_refl (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) |
| (A : Slots branch (m + 1) → Type u) |
| (values : Values 𝕜 repeats (.refl (m + 1)) A) |
| (exterior : CutTerminalSplit.ExteriorTape 𝕜 repeats (.refl (m + 1)) A) : |
| reconstruct 𝕜 repeats (.refl (m + 1)) A values exterior = values () := rfl |
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| theorem reconstruct_step (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) |
| (i : Fin (branch n)) (p : Path branch n (m + 1)) |
| (A : Slots branch (n + 1) → Type u) |
| (values : Values 𝕜 repeats (.step i p) A) |
| (exterior : CutTerminalSplit.ExteriorTape 𝕜 repeats (.step i p) A) : |
| reconstruct 𝕜 repeats (.step i p) A values exterior = |
| RecursiveSampler.combine 𝕜 A i (repeats (n + 1)) |
| (fun j => exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩) |
| (fun h => reconstruct 𝕜 repeats p (childFamily A i) |
| (fun terminal => values ((h, false), terminal)) |
| (CutSamplerReplayGrouping.subExterior 𝕜 repeats i p A (h, false) exterior)) |
| (fun h => reconstruct 𝕜 repeats p (childFamily A i) |
| (fun terminal => values ((h, true), terminal)) |
| (CutSamplerReplayGrouping.subExterior 𝕜 repeats i p A (h, true) exterior)) := rfl |
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| theorem reconstruct_splitTape (𝕜 : Type w) [Field 𝕜] |
| (repeats : Nat → Nat) (p : Path branch n (m + 1)) : |
| ∀ (A : Slots branch n → Type u) |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats p A), |
| reconstruct 𝕜 repeats p A |
| (fun terminal => UniformChildSum.recursiveSum (p.family A) |
| ((CutTerminalSplit.splitTape 𝕜 repeats p A tape).1 terminal)) |
| (CutTerminalSplit.splitTape 𝕜 repeats p A tape).2 = |
| CutSamplerRefinement.evaluate 𝕜 repeats p A tape := by |
| induction n generalizing m with |
| | zero => |
| have h := p.height_le |
| omega |
| | succ n ih => |
| cases p with |
| | refl => |
| intro A tape |
| rfl |
| | step i p => |
| intro A tape |
| rw [reconstruct_step, CutSamplerLocality.evaluate_step] |
| apply congrArg₂ (RecursiveSampler.combine 𝕜 A i (repeats (n + 1)) |
| (fun j => tape (.inl j))) |
| · funext h |
| rw [CutSamplerReplayGrouping.subExterior_splitTape] |
| simp_rw [CutTerminalZero.splitTape_terminal_step] |
| exact ih p (childFamily A i) |
| (CutSamplerLocality.subTape 𝕜 repeats i p A tape h false) |
| · funext h |
| rw [CutSamplerReplayGrouping.subExterior_splitTape] |
| simp_rw [CutTerminalZero.splitTape_terminal_step] |
| exact ih p (childFamily A i) |
| (CutSamplerLocality.subTape 𝕜 repeats i p A tape h true) |
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| theorem reconstruct_join (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) |
| (p : Path branch n (m + 1)) (A : Slots branch n → Type u) |
| (terminal : CutTerminalSplit.TerminalTape 𝕜 repeats p A) |
| (exterior : CutTerminalSplit.ExteriorTape 𝕜 repeats p A) : |
| reconstruct 𝕜 repeats p A |
| (fun call => UniformChildSum.recursiveSum (p.family A) (terminal call)) exterior = |
| CutSamplerRefinement.evaluate 𝕜 repeats p A |
| ((CutTerminalSplit.splitTape 𝕜 repeats p A).symm (terminal, exterior)) := by |
| simpa only [Equiv.apply_symm_apply] using |
| reconstruct_splitTape 𝕜 repeats p A |
| ((CutTerminalSplit.splitTape 𝕜 repeats p A).symm (terminal, exterior)) |
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| end |
| end PerfectCompleteness.LowerCutScalar |
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| end |
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| end OAI |
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