openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /CutTerminalZero.lean
| import OAI.Computability.PerfectCompleteness.Foundations.CutTerminalSplit | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerLocality | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutTerminalZero | |
| open RecursiveSpaces DescendantSpaces RecursiveSampler TerminalCalls | |
| open scoped Classical | |
| noncomputable section | |
| universe u w | |
| variable {branch : Nat → Nat} {n m : Nat} | |
| (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| theorem terminalFactorEquiv_refl (A : Slots branch (m + 1) → Type u) | |
| (terminal : TerminalIndex repeats (.refl (m + 1))) | |
| (x : CutSamplerRefinement.RefinedSpace 𝕜 repeats (.refl (m + 1)) A | |
| (drawIndex repeats (.refl (m + 1)) terminal)) : | |
| CutTerminalSplit.terminalFactorEquiv 𝕜 repeats (.refl (m + 1)) A terminal x = x := rfl | |
| theorem terminalFactorEquiv_step (i : Fin (branch n)) | |
| (p : Path branch n (m + 1)) (A : Slots branch (n + 1) → Type u) | |
| (call : Fin (repeats (n + 1)) × Bool) (terminal : TerminalIndex repeats p) | |
| (x : CutSamplerRefinement.RefinedSpace 𝕜 repeats p (childFamily A i) | |
| (drawIndex repeats p terminal)) : | |
| CutTerminalSplit.terminalFactorEquiv 𝕜 repeats (.step i p) A (call, terminal) x = | |
| CutTerminalSplit.terminalFactorEquiv 𝕜 repeats p (childFamily A i) terminal x := rfl | |
| theorem splitTape_terminal_step (i : Fin (branch n)) | |
| (p : Path branch n (m + 1)) (A : Slots branch (n + 1) → Type u) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats (.step i p) A) | |
| (h : Fin (repeats (n + 1))) (b : Bool) (terminal : TerminalIndex repeats p) : | |
| (CutTerminalSplit.splitTape 𝕜 repeats (.step i p) A tape).1 ((h, b), terminal) = | |
| (CutTerminalSplit.splitTape 𝕜 repeats p (childFamily A i) | |
| (CutSamplerLocality.subTape 𝕜 repeats i p A tape h b)).1 terminal := by | |
| simp only [CutTerminalSplit.splitTape_terminal_apply] | |
| rfl | |
| theorem zeroAtClean_iff (p : Path branch n (m + 1)) : | |
| ∀ (A : Slots branch n → Type u) (clean : Fin (branch m) → Prop) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats p A), | |
| (∀ terminal i, clean i → | |
| (CutTerminalSplit.splitTape 𝕜 repeats p A tape).1 terminal i = 0) ↔ | |
| CutSamplerLocality.ZeroAtClean 𝕜 repeats p A clean tape := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro A clean tape | |
| constructor | |
| · intro hzero i hi | |
| exact hzero () i hi | |
| · intro hzero terminal i hi | |
| cases terminal | |
| exact hzero i hi | |
| | step i p => | |
| intro A clean tape | |
| constructor | |
| · intro hzero h b | |
| apply (ih p (childFamily A i) clean | |
| (CutSamplerLocality.subTape 𝕜 repeats i p A tape h b)).mp | |
| intro terminal j hj | |
| apply Subtype.ext | |
| have hread := congrArg | |
| (fun f : CutTerminalSplit.TerminalFactor 𝕜 (.step i p) A => (f j).val) | |
| (splitTape_terminal_step 𝕜 repeats i p A tape h b terminal) | |
| exact hread.symm.trans (congrArg Subtype.val (hzero ((h, b), terminal) j hj)) | |
| · intro hzero terminal j hj | |
| rcases terminal with ⟨⟨h, b⟩, terminal⟩ | |
| have hsub := (ih p (childFamily A i) clean | |
| (CutSamplerLocality.subTape 𝕜 repeats i p A tape h b)).mpr (hzero h b) | |
| apply Subtype.ext | |
| have hread := congrArg | |
| (fun f : CutTerminalSplit.TerminalFactor 𝕜 (.step i p) A => (f j).val) | |
| (splitTape_terminal_step 𝕜 repeats i p A tape h b terminal) | |
| exact hread.trans (congrArg Subtype.val (hsub terminal j hj)) | |
| theorem evaluate_factors_of_split_zero (p : Path branch n (m + 1)) | |
| (A : Slots branch n → Type u) (clean : Fin (branch m) → Prop) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats p A) | |
| (hzero : ∀ terminal i, clean i → | |
| (CutTerminalSplit.splitTape 𝕜 repeats p A tape).1 terminal i = 0) : | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val ∈ | |
| FactoredFunctions.factoringSpace (CutSamplerLocality.view p A clean) := | |
| CutSamplerLocality.evaluate_factors 𝕜 repeats p A clean tape | |
| ((zeroAtClean_iff 𝕜 repeats p A clean tape).mp hzero) | |
| end | |
| end PerfectCompleteness.CutTerminalZero | |
| end | |
| end OAI | |