| import Mathlib |
| import OAI.Computability.VertexCover.PCP.DegreeReplacement |
| import OAI.Computability.VertexCover.PCP.ZigzagSpectral |
|
|
| namespace OAI |
|
|
| |
|
|
| open scoped BigOperators |
|
|
| namespace UniqueGames.Foundations.PCP.Overlay |
|
|
| open PoweringWalks SpectralReturn |
|
|
| variable {V D E A : Type*} |
|
|
| def originalPortGraph (G : ConstraintGraph V (V × D) A) : PortGraph V D where |
| rot := G.reverse |
| rot_involutive := G.reverse_involutive |
|
|
| def rotate (G : PortGraph V D) (H : PortGraph V E) : |
| V × (D ⊕ E) → V × (D ⊕ E) |
| | (v, Sum.inl d) => ((G.rot (v, d)).1, Sum.inl (G.rot (v, d)).2) |
| | (v, Sum.inr e) => ((H.rot (v, e)).1, Sum.inr (H.rot (v, e)).2) |
|
|
| theorem rotate_involutive (G : PortGraph V D) (H : PortGraph V E) : |
| Function.Involutive (rotate G H) := by |
| rintro ⟨v, p⟩ |
| rcases p with d | e |
| · exact congrArg (fun a : V × D => (a.1, (Sum.inl a.2 : D ⊕ E))) |
| (G.rot_involutive (v, d)) |
| · exact congrArg (fun a : V × E => (a.1, (Sum.inr a.2 : D ⊕ E))) |
| (H.rot_involutive (v, e)) |
|
|
| def portGraph (G : PortGraph V D) (H : PortGraph V E) : PortGraph V (D ⊕ E) where |
| rot := |
| { toFun := rotate G H |
| invFun := rotate G H |
| left_inv := rotate_involutive G H |
| right_inv := rotate_involutive G H } |
| rot_involutive := rotate_involutive G H |
|
|
| @[simp] theorem portGraph_rot_inl (G : PortGraph V D) (H : PortGraph V E) |
| (v : V) (d : D) : |
| (portGraph G H).rot (v, Sum.inl d) = |
| ((G.rot (v, d)).1, Sum.inl (G.rot (v, d)).2) := rfl |
|
|
| @[simp] theorem portGraph_rot_inr (G : PortGraph V D) (H : PortGraph V E) |
| (v : V) (e : E) : |
| (portGraph G H).rot (v, Sum.inr e) = |
| ((H.rot (v, e)).1, Sum.inr (H.rot (v, e)).2) := rfl |
|
|
| def constraintGraph (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) : |
| ConstraintGraph V (V × (D ⊕ E)) A where |
| reverse := (portGraph (originalPortGraph G) H).rot |
| reverse_involutive := (portGraph (originalPortGraph G) H).rot_involutive |
| tail := Prod.fst |
| accepts p a b := match p.2 with |
| | Sum.inl d => G.accepts (p.1, d) a b |
| | Sum.inr _ => true |
| reverse_accepts := by |
| rintro ⟨v, p⟩ a b |
| rcases p with d | e |
| · exact G.reverse_accepts (v, d) a b |
| · rfl |
|
|
| @[simp] theorem constraintGraph_tail (G : ConstraintGraph V (V × D) A) |
| (H : PortGraph V E) : (constraintGraph G H).tail = Prod.fst := rfl |
|
|
| @[simp] theorem constraintGraph_portGraph (G : ConstraintGraph V (V × D) A) |
| (H : PortGraph V E) : |
| originalPortGraph (constraintGraph G H) = portGraph (originalPortGraph G) H := rfl |
|
|
| @[simp] theorem edgeSatisfied_inl (G : ConstraintGraph V (V × D) A) |
| (H : PortGraph V E) (htail : G.tail = Prod.fst) |
| (labeling : V → A) (v : V) (d : D) : |
| (constraintGraph G H).edgeSatisfied labeling (v, Sum.inl d) = |
| G.edgeSatisfied labeling (v, d) := by |
| change G.accepts (v, d) (labeling v) (labeling (G.reverse (v, d)).1) = |
| G.accepts (v, d) (labeling (G.tail (v, d))) |
| (labeling (G.tail (G.reverse (v, d)))) |
| rw [htail] |
|
|
| @[simp] theorem edgeSatisfied_inr (G : ConstraintGraph V (V × D) A) |
| (H : PortGraph V E) (labeling : V → A) (v : V) (e : E) : |
| (constraintGraph G H).edgeSatisfied labeling (v, Sum.inr e) = true := rfl |
|
|
| theorem complete (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) |
| (htail : G.tail = Prod.fst) (labeling : V → A) |
| (h : ∀ a, G.edgeSatisfied labeling a = true) : |
| ∀ a, (constraintGraph G H).edgeSatisfied labeling a = true := by |
| rintro ⟨v, p⟩ |
| rcases p with d | e |
| · rw [edgeSatisfied_inl G H htail] |
| exact h (v, d) |
| · rfl |
|
|
| theorem satisfiable_iff (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) |
| (htail : G.tail = Prod.fst) : |
| (constraintGraph G H).Satisfiable ↔ G.Satisfiable := by |
| constructor |
| · rintro ⟨labeling, h⟩ |
| refine ⟨labeling, ?_⟩ |
| rintro ⟨v, d⟩ |
| rw [← edgeSatisfied_inl G H htail] |
| exact h (v, Sum.inl d) |
| · rintro ⟨labeling, h⟩ |
| exact ⟨labeling, complete G H htail labeling h⟩ |
|
|
| theorem rejectionCount_eq [Fintype V] [Fintype D] [Fintype E] |
| (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) |
| (htail : G.tail = Prod.fst) (labeling : V → A) : |
| (constraintGraph G H).rejectionCount labeling = G.rejectionCount labeling := by |
| classical |
| simp only [DegreeReplacement.rejectionCount_eq_sum, Fintype.sum_prod_type, |
| Fintype.sum_sum_type] |
| simp [edgeSatisfied_inl G H htail] |
|
|
| theorem card_ports [Fintype D] [Fintype E] : |
| Fintype.card (D ⊕ E) = Fintype.card D + Fintype.card E := Fintype.card_sum |
|
|
| theorem card_darts [Fintype V] [Fintype D] [Fintype E] : |
| Fintype.card (V × (D ⊕ E)) = |
| Fintype.card (V × D) + Fintype.card (V × E) := by |
| simp only [Fintype.card_prod, Fintype.card_sum, Nat.mul_add] |
|
|
| variable [Fintype V] [Fintype D] [Fintype E] |
|
|
| omit [Fintype V] in |
| |
| theorem operator_mix [Nonempty D] [Nonempty E] |
| (G : PortGraph V D) (H : PortGraph V E) (f : V → ℝ) (v : V) : |
| averagingOperator (portGraph G H) f v = |
| (Fintype.card D : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) * |
| averagingOperator G f v + |
| (Fintype.card E : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) * |
| averagingOperator H f v := by |
| have hd : (0 : ℝ) < Fintype.card D := by |
| exact_mod_cast (Fintype.card_pos : 0 < Fintype.card D) |
| have he : (0 : ℝ) < Fintype.card E := by |
| exact_mod_cast (Fintype.card_pos : 0 < Fintype.card E) |
| change mean (fun p : D ⊕ E => f ((portGraph G H).rot (v, p)).1) = _ |
| rw [mean_eq_sum_div_card] |
| simp only [Fintype.sum_sum_type, Fintype.card_sum, Nat.cast_add, |
| portGraph_rot_inl, portGraph_rot_inr] |
| unfold averagingOperator |
| rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card] |
| field_simp [hd.ne', he.ne', (add_pos hd he).ne'] |
|
|
| theorem energy_convex_mix (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) |
| (f g : V → ℝ) : |
| energy (fun v => a * f v + b * g v) ≤ a * energy f + b * energy g := by |
| calc |
| energy (fun v => a * f v + b * g v) ≤ |
| mean (fun v => a * (f v) ^ 2 + b * (g v) ^ 2) := by |
| apply mean_mono |
| intro v |
| have hnormalize := congrArg (fun t : ℝ => t * (a * (f v) ^ 2 + b * (g v) ^ 2)) hab |
| have hnonneg := mul_nonneg (mul_nonneg ha hb) (sq_nonneg (f v - g v)) |
| nlinarith |
| _ = a * energy f + b * energy g := by |
| rw [mean_add, mean_mul_left, mean_mul_left] |
| rfl |
|
|
| theorem energy_bound [Nonempty V] [Nonempty D] [Nonempty E] |
| (G : PortGraph V D) (H : PortGraph V E) (lambda : ℝ) |
| (certificate : SpectralCertificate H lambda) (f : V → ℝ) (hf : mean f = 0) : |
| energy (averagingOperator (portGraph G H) f) ≤ |
| ((Fintype.card D : ℝ) + (Fintype.card E : ℝ) * lambda ^ 2) / |
| ((Fintype.card D : ℝ) + Fintype.card E) * energy f := by |
| let a : ℝ := (Fintype.card D : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) |
| let b : ℝ := (Fintype.card E : ℝ) / ((Fintype.card D : ℝ) + Fintype.card E) |
| have hd : (0 : ℝ) < Fintype.card D := by |
| exact_mod_cast (Fintype.card_pos : 0 < Fintype.card D) |
| have he : (0 : ℝ) < Fintype.card E := by |
| exact_mod_cast (Fintype.card_pos : 0 < Fintype.card E) |
| have hden : (0 : ℝ) < (Fintype.card D : ℝ) + Fintype.card E := add_pos hd he |
| have ha : 0 ≤ a := div_nonneg hd.le hden.le |
| have hb : 0 ≤ b := div_nonneg he.le hden.le |
| have hab : a + b = 1 := by |
| dsimp [a, b] |
| rw [← add_div, div_self hden.ne'] |
| have hop : averagingOperator (portGraph G H) f = |
| fun v => a * averagingOperator G f v + b * averagingOperator H f v := by |
| funext v |
| exact operator_mix G H f v |
| calc |
| energy (averagingOperator (portGraph G H) f) ≤ |
| a * energy (averagingOperator G f) + b * energy (averagingOperator H f) := by |
| rw [hop] |
| exact energy_convex_mix a b ha hb hab _ _ |
| _ ≤ a * energy f + b * (lambda ^ 2 * energy f) := |
| add_le_add |
| (mul_le_mul_of_nonneg_left (ZigzagSpectral.averaging_energy_le G f) ha) |
| (mul_le_mul_of_nonneg_left (certificate.contraction f hf) hb) |
| _ = _ := by |
| dsimp [a, b] |
| ring |
|
|
| theorem energy_bound_three_quarters [Nonempty V] |
| (G : PortGraph V D) (H : PortGraph V E) |
| (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E) |
| (certificate : SpectralCertificate H (1 / 2 : ℝ)) |
| (f : V → ℝ) (hf : mean f = 0) : |
| energy (averagingOperator (portGraph G H) f) ≤ (3 / 4 : ℝ) * energy f := by |
| let _ : Nonempty E := Fintype.card_pos_iff.mp (lt_of_lt_of_le Nat.zero_lt_one hpositive) |
| let _ : Nonempty D := Fintype.card_pos_iff.mp (by omega) |
| have he : (1 : ℝ) ≤ Fintype.card E := by exact_mod_cast hpositive |
| have hd : (Fintype.card D : ℝ) = (Fintype.card E : ℝ) + 1 := by exact_mod_cast hdegree |
| have hden : (0 : ℝ) < (Fintype.card D : ℝ) + Fintype.card E := by linarith |
| have hcoefficient : |
| ((Fintype.card D : ℝ) + (Fintype.card E : ℝ) * (1 / 2 : ℝ) ^ 2) / |
| ((Fintype.card D : ℝ) + Fintype.card E) ≤ (3 / 4 : ℝ) := by |
| apply (div_le_iff₀ hden).mpr |
| nlinarith |
| exact (energy_bound G H (1 / 2) certificate f hf).trans |
| (mul_le_mul_of_nonneg_right hcoefficient (energy_nonnegative f)) |
|
|
| theorem spectralCertificate_seven_eighths [Nonempty V] |
| (G : PortGraph V D) (H : PortGraph V E) |
| (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E) |
| (certificate : SpectralCertificate H (1 / 2 : ℝ)) : |
| SpectralCertificate (portGraph G H) (7 / 8 : ℝ) where |
| nonnegative := by norm_num |
| lt_one := by norm_num |
| contraction := by |
| intro f hf |
| exact (energy_bound_three_quarters G H hdegree hpositive certificate f hf).trans |
| (mul_le_mul_of_nonneg_right (by norm_num : (3 / 4 : ℝ) ≤ (7 / 8 : ℝ) ^ 2) |
| (energy_nonnegative f)) |
|
|
| theorem constraintGraph_spectralCertificate [Nonempty V] |
| (G : ConstraintGraph V (V × D) A) (H : PortGraph V E) |
| (hdegree : Fintype.card D = Fintype.card E + 1) (hpositive : 1 ≤ Fintype.card E) |
| (certificate : SpectralCertificate H (1 / 2 : ℝ)) : |
| SpectralCertificate (originalPortGraph (constraintGraph G H)) (7 / 8 : ℝ) := by |
| rw [constraintGraph_portGraph] |
| exact spectralCertificate_seven_eighths (originalPortGraph G) H |
| hdegree hpositive certificate |
|
|
| end UniqueGames.Foundations.PCP.Overlay |
|
|
| end OAI |
|
|
