openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /CutSamplerLocality.lean
| import OAI.Computability.PerfectCompleteness.Construction.FactoredFunctionsLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerRefinement | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutSamplerLocality | |
| open scoped BigOperators Classical | |
| open RecursiveSpaces DescendantSpaces RecursiveSampler PointwiseSpaces | |
| open CutSamplerRefinement | |
| noncomputable section | |
| universe u w | |
| variable {branch : Nat → Nat} {n m : Nat} | |
| def View : {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (A : Slots branch n → Type u) → (clean : Fin (branch m) → Prop) → Type u | |
| | _, m, .refl _, A, clean => | |
| (i : {i : Fin (branch m) // ¬ clean i}) → Assignment (childFamily A i.val) | |
| | _, _, .step i p, A, clean => | |
| ((j : OffPath i) → Assignment (childFamily A j.val)) × | |
| View p (childFamily A i) clean | |
| def view : {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (A : Slots branch n → Type u) → (clean : Fin (branch m) → Prop) → | |
| Assignment A → View p A clean | |
| | _, _, .refl _, A, _, x => fun i => childRestriction A i.val x | |
| | _, _, .step i p, A, clean, x => | |
| (fun j => childRestriction A j.val x, | |
| view p (childFamily A i) clean (childRestriction A i x)) | |
| def subTape (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (i : Fin (branch n)) (p : Path branch n m) | |
| (A : Slots branch (n + 1) → Type u) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats (.step i p) A) | |
| (h : Fin (repeats (n + 1))) (b : Bool) : | |
| CutSamplerRefinement.Tape 𝕜 repeats p (childFamily A i) := | |
| fun j => tape (Sum.inr ((h, b), j)) | |
| def ZeroAtClean (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → (A : Slots branch n → Type u) → | |
| (clean : Fin (branch m) → Prop) → CutSamplerRefinement.Tape 𝕜 repeats p A → Prop | |
| | _, _, .refl _, _, clean, tape => ∀ i, clean i → tape () i = 0 | |
| | _ + 1, _, .step i p, A, clean, tape => | |
| ∀ h b, ZeroAtClean 𝕜 repeats p (childFamily A i) clean | |
| (subTape 𝕜 repeats i p A tape h b) | |
| theorem evaluate_refl (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (A : Slots branch (m + 1) → Type u) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats (.refl (m + 1)) A) : | |
| CutSamplerRefinement.evaluate 𝕜 repeats (.refl (m + 1)) A tape = | |
| UniformChildSum.recursiveSum A (tape ()) := rfl | |
| theorem evaluate_step (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) | |
| (i : Fin (branch n)) (p : Path branch n m) | |
| (A : Slots branch (n + 1) → Type u) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats (.step i p) A) : | |
| CutSamplerRefinement.evaluate 𝕜 repeats (.step i p) A tape = | |
| RecursiveSampler.combine 𝕜 A i (repeats (n + 1)) | |
| (fun j => tape (Sum.inl j)) | |
| (fun h => CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h false)) | |
| (fun h => CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h true)) := rfl | |
| theorem evaluate_eq_of_view_eq (𝕜 : Type w) [Field 𝕜] | |
| (repeats : Nat → Nat) (p : Path branch n (m + 1)) : | |
| ∀ (A : Slots branch n → Type u) (clean : Fin (branch m) → Prop) | |
| (tape : CutSamplerRefinement.Tape 𝕜 repeats p A), | |
| ZeroAtClean 𝕜 repeats p A clean tape → | |
| ∀ x y : Assignment A, view p A clean x = view p A clean y → | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val x = | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val y := by | |
| induction n generalizing m with | |
| | zero => | |
| have h := p.height_le | |
| omega | |
| | succ n ih => | |
| cases p with | |
| | refl => | |
| intro A clean tape hzero x y hview | |
| have heval (z : Assignment A) : | |
| (UniformChildSum.recursiveSum (𝕜 := 𝕜) A (tape ())).val z = | |
| ∑ i : Fin (branch n), (tape () i).val (childRestriction A i z) := by | |
| have hsum := congrFun | |
| (UniformChildSum.childSum_apply (𝕜 := 𝕜) (childRestriction A) | |
| (fun i => space 𝕜 branch n (childFamily A i)) (tape ())) z | |
| simp only [Finset.sum_apply, PointwiseSpaces.pullback_apply] at hsum | |
| refine hsum.trans ?_ | |
| refine Finset.sum_congr (ι := Fin (branch n)) ?_ ?_ | |
| · ext i | |
| simp only [Finset.mem_univ] | |
| · intro i _ | |
| rfl | |
| rw [evaluate_refl, heval x, heval y] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| by_cases hi : clean i | |
| · rw [hzero i hi] | |
| rfl | |
| · exact congrArg (tape () i).val (congrFun hview ⟨i, hi⟩) | |
| | step i p => | |
| intro A clean tape hzero x y hview | |
| have hout : ∀ j : OffPath i, childRestriction A j.val x = childRestriction A j.val y := | |
| congrFun (congrArg Prod.fst hview) | |
| have hin : view p (childFamily A i) clean (childRestriction A i x) = | |
| view p (childFamily A i) clean (childRestriction A i y) := | |
| congrArg Prod.snd hview | |
| rw [evaluate_step] | |
| simp only [RecursiveSampler.combine, RecursiveSampler.combineFunction, | |
| Finset.sum_apply, Pi.add_apply, Pi.mul_apply, PointwiseSpaces.pullback_apply] | |
| change | |
| (∑ j : OffPath i, (tape (Sum.inl j)).val (childRestriction A j.val x)) + | |
| (∑ h, (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h false)).val (childRestriction A i x) * | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h true)).val (childRestriction A i x)) = | |
| (∑ j : OffPath i, (tape (Sum.inl j)).val (childRestriction A j.val y)) + | |
| (∑ h, (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h false)).val (childRestriction A i y) * | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i) | |
| (subTape 𝕜 repeats i p A tape h true)).val (childRestriction A i y)) | |
| congr 1 | |
| · exact Finset.sum_congr rfl (fun j _ => congrArg (tape (Sum.inl j)).val (hout j)) | |
| · apply Finset.sum_congr rfl | |
| intro h _ | |
| rw [ih p (childFamily A i) clean (subTape 𝕜 repeats i p A tape h false) | |
| (hzero h false) _ _ hin, | |
| ih p (childFamily A i) clean (subTape 𝕜 repeats i p A tape h true) | |
| (hzero h true) _ _ hin] | |
| theorem evaluate_factors (𝕜 : Type w) [Field 𝕜] | |
| (repeats : Nat → Nat) (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 : ZeroAtClean 𝕜 repeats p A clean tape) : | |
| (CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val ∈ | |
| FactoredFunctions.factoringSpace (view p A clean) := | |
| evaluate_eq_of_view_eq 𝕜 repeats p A clean tape hzero | |
| end | |
| end PerfectCompleteness.CutSamplerLocality | |
| end | |
| end OAI | |