openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /ChildFiniteCallCollisionTransfer.lean
| import OAI.Computability.PerfectCompleteness.Algebra.ChildBilinearCollisionTransfer | |
| import OAI.Computability.PerfectCompleteness.Sampling.ChildCallNumberingLaws | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.ChildFiniteCallCollisionTransfer | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces TreeSourceSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C] | |
| {Z : Fin (branch n) → Type*} [∀ i, Fintype (Z i)] | |
| theorem marked_restricted_le_uniform_full | |
| (rows : Nat → Nat) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (projected : (i : Fin (branch n)) → Z i → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ i z s k, MixedSupport.Projection | |
| (childSlots slots i s k) (projected i z s k)) | |
| (choiceLaw : (i : Fin (branch n)) → FiniteDistribution (Z i)) | |
| (β : ℝ) (hβ : 0 ≤ β) (hβ' : β < 1) | |
| (F₀ F₁ : CutChildGrouping.Raw (C := C) slots rows → | |
| Option (ChildBilinearCollisionTransfer.ParentForm slots)) : | |
| (SparseReplacement.markedLaw | |
| (fun i => FiniteDistribution.uniform | |
| (ChildCallNumbering.Raw (C := C) rows (childSlots slots i))) | |
| (ChildCallNumberingLaws.replacementLaws (C := C) rows (childSlots slots) | |
| projected p choiceLaw) β hβ hβ'.le).probability | |
| (fun z => BilinearCollisionTransfer.restrictedCollision | |
| (F₀ z.2) (F₁ z.2) (ChildUnmarkedSpace.space slots z.1)) ≤ | |
| (FiniteProduct.law (fun i => FiniteDistribution.uniform | |
| (ChildCallNumbering.Raw (C := C) rows (childSlots slots i)))).probability | |
| (fun x => BilinearCollisionTransfer.fullCollision (F₀ x) (F₁ x)) + | |
| 2 * (β * (ChildBlockCardinality.bound branch n t (Fintype.card C) rows : ℝ) / | |
| (1 - β)) + | |
| Real.sqrt ((1 + β ^ 2 * | |
| ((ChildBlockCardinality.bound branch n t (Fintype.card C) rows : ℝ) - 1)) ^ | |
| branch n - 1) / 2 := by | |
| have h := ChildBilinearCollisionTransfer.marked_restricted_le_uniform_full | |
| (Fintype.card C) rows slots projected p choiceLaw β hβ hβ' | |
| (fun x => F₀ (ChildCallNumberingLaws.unnumber (C := C) rows (childSlots slots) x)) | |
| (fun x => F₁ (ChildCallNumberingLaws.unnumber (C := C) rows (childSlots slots) x)) | |
| rw [← ChildCallNumberingLaws.markedLaw_numbered (C := C) rows (childSlots slots) | |
| projected p choiceLaw β hβ hβ'.le, FiniteDistribution.probability_pushforward] at h | |
| rw [← ChildCallNumberingLaws.uniform_product_numbered (C := C) rows (childSlots slots), | |
| FiniteDistribution.probability_pushforward] at h | |
| simpa only [ChildCallNumberingLaws.unnumber_number] using h | |
| end | |
| end PerfectCompleteness.ChildFiniteCallCollisionTransfer | |
| end | |
| end OAI | |