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import OAI.Computability.VertexCover.PCP.DegreeReplacement
import OAI.Computability.VertexCover.PCP.ExpanderFamily
namespace OAI
namespace UniqueGames.Foundations.PCP.CloudPadding
open scoped BigOperators
open DegreeReplacement
variable {V E A : Type*}
def paddedSize (k : Nat) : Nat := if k = 0 then 0 else ExpanderFamily.size k
@[simp] theorem paddedSize_zero : paddedSize 0 = 0 := rfl
theorem paddedSize_of_pos {k : Nat} (hk : 0 < k) :
paddedSize k = ExpanderFamily.size k := by
simp only [paddedSize, ite_eq_right (Nat.ne_of_gt hk)]
theorem le_paddedSize (k : Nat) : k ≤ paddedSize k := by
by_cases hk : k = 0
· subst k
simp
· rw [paddedSize, ite_eq_right hk]
exact ExpanderFamily.le_size k
theorem paddedSize_le_mul (k : Nat) : paddedSize k ≤ ExpanderFamily.growth * k := by
by_cases hk : k = 0
· subst k
simp
· rw [paddedSize, ite_eq_right hk]
exact ExpanderFamily.size_le_mul (Nat.pos_of_ne_zero hk)
def paddedCloudEquiv (G : ConstraintGraph V E A) (dummy : V → Type*) (v : V) :
Cloud (paddedGraph G dummy) v ≃ Cloud G v ⊕ dummy v where
toFun := by
rintro ⟨e, he⟩
rcases e with e | ⟨w, d⟩
· exact Sum.inl ⟨e, he⟩
· change w = v at he
cases he
exact Sum.inr d
invFun := fun z => match z with
| Sum.inl e => ⟨Sum.inl e.val, e.property⟩
| Sum.inr d => ⟨Sum.inr ⟨v, d⟩, rfl⟩
left_inv := by
rintro ⟨e, he⟩
rcases e with e | ⟨w, d⟩
· rfl
· change w = v at he
cases he
rfl
right_inv := by
intro z
cases z <;> rfl
def dartCloudEquiv (G : ConstraintGraph V E A) : E ≃ Σ v : V, Cloud G v where
toFun e := ⟨G.tail e, ⟨e, rfl⟩⟩
invFun z := z.2.val
left_inv _ := rfl
right_inv := by
rintro ⟨v, ⟨e, he⟩⟩
cases he
rfl
variable [Fintype V] [DecidableEq V] [Fintype E]
theorem sum_card_cloud (G : ConstraintGraph V E A) :
(∑ v, Fintype.card (Cloud G v)) = Fintype.card E := by
simpa only [Fintype.card_sigma] using (Fintype.card_congr (dartCloudEquiv G)).symm
theorem card_padded_cloud (G : ConstraintGraph V E A) (dummy : V → Type*)
[∀ v, Fintype (dummy v)] (v : V) :
Fintype.card (Cloud (paddedGraph G dummy) v) =
Fintype.card (Cloud G v) + Fintype.card (dummy v) := by
simpa only [Fintype.card_sum] using Fintype.card_congr (paddedCloudEquiv G dummy v)
abbrev dummy (G : ConstraintGraph V E A) (v : V) :=
Fin (paddedSize (Fintype.card (Cloud G v)) - Fintype.card (Cloud G v))
theorem card_cloud (G : ConstraintGraph V E A) (v : V) :
Fintype.card (Cloud (paddedGraph G (dummy G)) v) =
paddedSize (Fintype.card (Cloud G v)) := by
rw [card_padded_cloud]
simp only [dummy, Fintype.card_fin]
exact Nat.add_sub_of_le (le_paddedSize _)
omit [Fintype V] in
theorem card_dummy_of_empty (G : ConstraintGraph V E A) (v : V)
(hk : Fintype.card (Cloud G v) = 0) : Fintype.card (dummy G v) = 0 := by
simp [dummy, hk]
theorem card_cloud_of_empty (G : ConstraintGraph V E A) (v : V)
(hk : Fintype.card (Cloud G v) = 0) :
Fintype.card (Cloud (paddedGraph G (dummy G)) v) = 0 := by
rw [card_cloud, hk, paddedSize_zero]
theorem card_cloud_eq_family (G : ConstraintGraph V E A) (v : V)
(hk : 0 < Fintype.card (Cloud G v)) :
Fintype.card (Cloud (paddedGraph G (dummy G)) v) =
Fintype.card (ExpanderFamily.Vertex
(ExpanderFamily.level (Fintype.card (Cloud G v)))) := by
rw [card_cloud, paddedSize_of_pos hk, ExpanderFamily.card_vertex]
rfl
theorem card_paddedDart_eq_sum (G : ConstraintGraph V E A) :
Fintype.card (PaddedDart G (dummy G)) =
∑ v, paddedSize (Fintype.card (Cloud G v)) := by
calc
Fintype.card (PaddedDart G (dummy G)) =
∑ v, Fintype.card (Cloud (paddedGraph G (dummy G)) v) :=
(sum_card_cloud (paddedGraph G (dummy G))).symm
_ = ∑ v, paddedSize (Fintype.card (Cloud G v)) := by
apply Finset.sum_congr rfl
intro v _
exact card_cloud G v
theorem card_paddedDart_le (G : ConstraintGraph V E A) :
Fintype.card (PaddedDart G (dummy G)) ≤ ExpanderFamily.growth * Fintype.card E := by
rw [card_paddedDart_eq_sum]
calc
(∑ v, paddedSize (Fintype.card (Cloud G v))) ≤
∑ v, ExpanderFamily.growth * Fintype.card (Cloud G v) := by
apply Finset.sum_le_sum
intro v _
exact paddedSize_le_mul _
_ = ExpanderFamily.growth * Fintype.card E := by
rw [← Finset.mul_sum, sum_card_cloud]
end UniqueGames.Foundations.PCP.CloudPadding
end OAI
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