| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Expansion.DegreeReplacement |
| import OAI.Combinatorics.IndependentSets.Expansion.CloudRounding |
|
|
| namespace OAI |
|
|
| namespace IndependentSetsGames.Foundations.PCP.CloudRounding |
| open scoped BigOperators |
| section GraphRounding |
|
|
| open DegreeReplacement PoweringWalks |
|
|
| variable {V E A D : Type*} [Fintype V] [Fintype E] [Fintype A] [Fintype D] |
| variable [DecidableEq V] [DecidableEq E] [DecidableEq A] |
|
|
| def mismatchSet (G : ConstraintGraph V E A) (ell : E → A) (rounded : V → A) : |
| Finset E := |
| Finset.univ.filter (fun e => ell e ≠ rounded (G.tail e)) |
|
|
| def copiedRejectionCount (G : ConstraintGraph V E A) (ell : E → A) : Nat := |
| (Finset.univ.filter (fun e => G.accepts e (ell e) (ell (G.reverse e)) = false)).card |
|
|
| def innerDisagreementSet (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) : Finset (E × D) := |
| Finset.univ.filter (fun ed => ell ed.1 ≠ ell (innerRotation G H ed).1) |
|
|
| noncomputable def roundLabels [Nonempty A] |
| (G : ConstraintGraph V E A) (ell : E → A) : V → A := |
| fun v => majority (fun e : Cloud G v => ell e.val) |
|
|
| def mismatchFiberEquiv (G : ConstraintGraph V E A) (ell : E → A) |
| (rounded : V → A) : |
| (Σ v, ↥(minoritySet (fun e : Cloud G v => ell e.val) (rounded v))) ≃ |
| ↥(mismatchSet G ell rounded) where |
| toFun := by |
| rintro ⟨v, ⟨⟨e, he⟩, hbad⟩⟩ |
| cases he |
| exact ⟨e, by simpa [mismatchSet, minoritySet] using hbad⟩ |
| invFun e := |
| ⟨G.tail e.val, ⟨⟨e.val, rfl⟩, by |
| simpa [mismatchSet, minoritySet] using e.property⟩⟩ |
| left_inv := by |
| rintro ⟨v, ⟨⟨e, he⟩, hbad⟩⟩ |
| cases he |
| rfl |
| right_inv := by intro e; apply Subtype.ext; rfl |
|
|
| omit [Fintype A] [DecidableEq E] in |
| theorem mismatch_card_eq_sum (G : ConstraintGraph V E A) (ell : E → A) |
| (rounded : V → A) : |
| (mismatchSet G ell rounded).card = |
| ∑ v, (minoritySet (fun e : Cloud G v => ell e.val) (rounded v)).card := by |
| have h := Fintype.card_congr (mismatchFiberEquiv G ell rounded) |
| simpa only [Fintype.card_sigma, Fintype.card_coe] using h.symm |
|
|
| def disagreementFiberEquiv (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) : |
| (Σ v, ↥(disagreementSet (fun e : Cloud G v => ell e.val) |
| (fun ed => ((H v).rot ed).1))) ≃ ↥(innerDisagreementSet G H ell) where |
| toFun := by |
| rintro ⟨v, ⟨⟨⟨e, he⟩, d⟩, hbad⟩⟩ |
| cases he |
| refine ⟨(e, d), ?_⟩ |
| simpa [disagreementSet, innerDisagreementSet, innerRotation, |
| cloudIndexEquiv, cloudRotation] using hbad |
| invFun := by |
| rintro ⟨⟨e, d⟩, hbad⟩ |
| refine ⟨G.tail e, ⟨(⟨e, rfl⟩, d), ?_⟩⟩ |
| simpa [disagreementSet, innerDisagreementSet, innerRotation, |
| cloudIndexEquiv, cloudRotation] using hbad |
| left_inv := by |
| rintro ⟨v, ⟨⟨⟨e, he⟩, d⟩, hbad⟩⟩ |
| cases he |
| rfl |
| right_inv := by intro ed; apply Subtype.ext; rfl |
|
|
| omit [Fintype A] [DecidableEq E] in |
| theorem disagreement_card_eq_sum (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) : |
| (innerDisagreementSet G H ell).card = |
| ∑ v, (disagreementSet (fun e : Cloud G v => ell e.val) |
| (fun ed => ((H v).rot ed).1)).card := by |
| have h := Fintype.card_congr (disagreementFiberEquiv G H ell) |
| simpa only [Fintype.card_sigma, Fintype.card_coe] using h.symm |
|
|
| theorem rounded_mismatch_expansion [Nonempty A] |
| (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D) |
| (ell : E → A) (h : ℚ) |
| (expansion : ∀ v (S : Finset (Cloud G v)), |
| S.card ≤ Fintype.card (Cloud G v) / 2 → |
| h * (S.card : ℚ) ≤ ((directedCut (fun ed => ((H v).rot ed).1) S).card : ℚ)) : |
| h * ((mismatchSet G ell (roundLabels G ell)).card : ℚ) ≤ |
| ((innerDisagreementSet G H ell).card : ℚ) := by |
| rw [mismatch_card_eq_sum, disagreement_card_eq_sum, Nat.cast_sum, Nat.cast_sum, |
| Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro v _ |
| exact minority_expansion_bound (fun e : Cloud G v => ell e.val) |
| (fun ed => ((H v).rot ed).1) (roundLabels G ell v) h |
| (majority_maximal (fun e : Cloud G v => ell e.val)) (expansion v) |
|
|
| omit [Fintype V] [Fintype A] [DecidableEq V] [DecidableEq E] in |
| theorem rounded_endpoint_bound (G : ConstraintGraph V E A) |
| (ell : E → A) (rounded : V → A) : |
| G.rejectionCount rounded ≤ copiedRejectionCount G ell + |
| 2 * (mismatchSet G ell rounded).card := |
| dart_rejection_count_le G.tail G.reverse G.accepts ell rounded |
|
|
| omit [Fintype V] [Fintype E] [Fintype A] [Fintype D] [DecidableEq V] [DecidableEq E] in |
| @[simp] theorem copied_satisfied_inl (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) (e : E) (d : D) : |
| (replacementGraph G H).edgeSatisfied ell (e, Sum.inl d) = |
| decide (ell e = ell (innerRotation G H (e, d)).1) := rfl |
|
|
| omit [Fintype V] [Fintype E] [Fintype A] [Fintype D] [DecidableEq V] [DecidableEq E] in |
| @[simp] theorem copied_satisfied_inr (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) (e : E) (u : Unit) : |
| (replacementGraph G H).edgeSatisfied ell (e, Sum.inr u) = |
| G.accepts e (ell e) (ell (G.reverse e)) := rfl |
|
|
| omit [Fintype V] [Fintype A] [DecidableEq V] [DecidableEq E] in |
| theorem replacement_rejection_split (G : ConstraintGraph V E A) |
| (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) : |
| (replacementGraph G H).rejectionCount ell = |
| copiedRejectionCount G ell + (innerDisagreementSet G H ell).card := by |
| classical |
| simp only [rejectionCount_eq_sum, Fintype.sum_prod_type, Fintype.sum_sum_type, |
| copied_satisfied_inl, copied_satisfied_inr, decide_eq_false_iff_not, |
| Fintype.sum_unique, Finset.sum_add_distrib] |
| simp only [copiedRejectionCount, innerDisagreementSet, Finset.card_eq_sum_ones, |
| Finset.sum_filter, Fintype.sum_prod_type] |
| exact Nat.add_comm _ _ |
|
|
| theorem replacement_soundness [Nonempty A] |
| (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D) |
| (expansion : ∀ v (S : Finset (Cloud G v)), |
| S.card ≤ Fintype.card (Cloud G v) / 2 → |
| 2 * S.card ≤ (directedCut (fun ed => ((H v).rot ed).1) S).card) |
| (ell : E → A) : |
| G.rejectionCount (roundLabels G ell) ≤ (replacementGraph G H).rejectionCount ell := by |
| have hc := rounded_mismatch_expansion G H ell 2 (by |
| intro v S hS |
| exact_mod_cast expansion v S hS) |
| have hcNat : 2 * (mismatchSet G ell (roundLabels G ell)).card ≤ |
| (innerDisagreementSet G H ell).card := by exact_mod_cast hc |
| rw [replacement_rejection_split] |
| exact (rounded_endpoint_bound G ell (roundLabels G ell)).trans |
| (Nat.add_le_add_left hcNat _) |
|
|
| theorem padded_replacement_soundness [Nonempty A] |
| (G : ConstraintGraph V E A) (dummy : V → Type*) [∀ v, Fintype (dummy v)] |
| [∀ v, DecidableEq (dummy v)] |
| (H : ∀ v, PortGraph (Cloud (paddedGraph G dummy) v) D) |
| (expansion : ∀ v (S : Finset (Cloud (paddedGraph G dummy) v)), |
| S.card ≤ Fintype.card (Cloud (paddedGraph G dummy) v) / 2 → |
| 2 * S.card ≤ (directedCut (fun ed => ((H v).rot ed).1) S).card) |
| (ell : PaddedDart G dummy → A) : |
| G.rejectionCount (roundLabels (paddedGraph G dummy) ell) ≤ |
| (paddedReplacementGraph G dummy H).rejectionCount ell := by |
| simpa only [paddedReplacementGraph, padded_rejectionCount] using |
| replacement_soundness (paddedGraph G dummy) H expansion ell |
|
|
| end GraphRounding |
|
|
| theorem combine_counts (R b M c : Nat) (h : ℚ) (hh : 0 < h) |
| (endpoint : R ≤ b + 2 * M) (cloud : h * (M : ℚ) ≤ (c : ℚ)) : |
| (R : ℚ) ≤ max 1 (2 / h) * ((b : ℚ) + (c : ℚ)) := by |
| let K : ℚ := max 1 (2 / h) |
| have hK : 1 ≤ K := le_max_left _ _ |
| have hK0 : 0 ≤ K := le_trans zero_le_one hK |
| have hcoef : 2 ≤ K * h := (div_le_iff₀ hh).mp (le_max_right _ _) |
| have hm : (2 : ℚ) * M ≤ K * c := by |
| calc |
| (2 : ℚ) * M ≤ (K * h) * M := |
| mul_le_mul_of_nonneg_right hcoef (Nat.cast_nonneg M) |
| _ = K * (h * M) := mul_assoc _ _ _ |
| _ ≤ K * c := mul_le_mul_of_nonneg_left cloud hK0 |
| have hb : (b : ℚ) ≤ K * b := by |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right hK (Nat.cast_nonneg b) |
| have hr : (R : ℚ) ≤ (b : ℚ) + 2 * (M : ℚ) := by exact_mod_cast endpoint |
| exact hr.trans ((add_le_add hb hm).trans_eq (mul_add K (b : ℚ) (c : ℚ)).symm) |
|
|
| end IndependentSetsGames.Foundations.PCP.CloudRounding |
|
|
| end OAI |
|
|
