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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
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