| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Geometry.Witness |
| import OAI.Combinatorics.IndependentSets.Reduction.ThresholdDecidable |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets.ExplicitEnum |
|
|
| def mixed {A B : Type*} [DecidableEq A] [DecidableEq B] |
| (ea : ExplicitEnum A) (eb : ExplicitEnum B) (n i : ℕ) : ExplicitEnum (MixedTuple A B n i) := |
| subtype (pi (fin n) (fun _ => sum eb ea)) (fun x => ∀ h, (x h).isLeft = decide (h.val < i)) |
|
|
| def question {U V : Type*} [DecidableEq U] [DecidableEq V] |
| (eu : ExplicitEnum U) (ev : ExplicitEnum V) (n : ℕ) : ExplicitEnum (LayerQuestion U V n) := |
| sigma (fin (n+1)) (fun i => mixed eu ev n i.val) |
|
|
| def grid {U V L R : Type} [Fintype L] [Fintype R] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| (eu : ExplicitEnum U) (ev : ExplicitEnum V) (el : ExplicitEnum L) (er : ExplicitEnum R) |
| (n D : ℕ) [NeZero D] : |
| ExplicitEnum (GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=n)) D) := |
| sigma (question eu ev n) (fun q => pi (mixed el er n q.1.val) (fun _ => zmod D)) |
|
|
| def equiv {α : Type*} [DecidableEq α] (e : ExplicitEnum α) : Fin e.values.length ≃ α := |
| enumerationEquiv e.values e.nodup e.covers |
|
|
| lemma length_eq_card {α : Type*} [DecidableEq α] [Fintype α] (e : ExplicitEnum α) : |
| e.values.length = Fintype.card α := by |
| simpa using Fintype.card_congr e.equiv |
|
|
| end LargeIndependentSets.ExplicitEnum |
|
|
| namespace LargeIndependentSets |
| open BooleanJunta PhaseTest |
| open scoped BigOperators ENNReal |
|
|
| def rotationCode {k : ℕ} (x : Cube k) : Fin (2^k) := ⟨gridIndex x,gridIndex_lt x⟩ |
|
|
| def sampleLocationCode {U V L R C : Type} [Fintype L] [Fintype R] |
| {n s m k : ℕ} (lc : LabelCoverData U V L R C) (chain : Fin n → C) (hs : 0<s) |
| (o : SamplerOutcome L R n s m k) : |
| GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=n)) (m*2^k) := |
| let b : o.1.val := ⟨o.1.val.min' (supported_nonempty hs o.1), |
| o.1.val.min'_mem (supported_nonempty hs o.1)⟩ |
| ⟨occurrenceQuestion lc chain b.val, |
| fun a => ∑ j : o.1.val, ((o.2.1 j).val * |
| (rotationCode (o.2.2 ⟨j,chainComposite lc chain b.val.val j.val.val |
| (Finset.min'_le _ _ j.property) a⟩)).val : ℕ)⟩ |
|
|
| lemma sampleLocationCode_eq {U V L R C : Type} [Fintype L] [Fintype R] |
| {n s m k : ℕ} (lc : LabelCoverData U V L R C) (chain : Fin n → C) (hs : 0<s) |
| (o : SamplerOutcome L R n s m k) : |
| sampleLocationCode lc chain hs o = sampleLocation lc chain hs o := rfl |
|
|
| instance layeredImposedDecidable {U V L R C : Type} |
| [Fintype U] [Fintype V] [Fintype L] [Fintype R] [Fintype C] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| (lc : LabelCoverData U V L R C) (n : ℕ) (q q' : LayerQuestion U V n) |
| (π : LayerAnswer L R q → LayerAnswer L R q') : |
| Decidable ((layeredSystem lc n).imposed q q' π) := |
| LayerImposed.imposed_decidable lc n q q' π |
|
|
| namespace SamplerParameters |
|
|
| def locationCode (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (v : p.Vertices L R C) := |
| sampleLocationCode lc v.1 p.hs v.2.1 |
|
|
| lemma locationCode_eq (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (v : p.Vertices L R C) : |
| p.locationCode lc v = p.location lc v := rfl |
|
|
| def chainEnumeration (n : ℕ) {C O : Type} [DecidableEq C] [DecidableEq O] |
| (ec : ExplicitEnum C) (eo : ExplicitEnum O) : |
| ExplicitEnum ((Fin n → C) × O) := |
| ExplicitEnum.prod (ExplicitEnum.pi (ExplicitEnum.fin n) (fun _ => ec)) eo |
|
|
| def graphAdjDecidable (p : SamplerParameters) {U V L R C : Type} |
| [Fintype U] [Fintype V] [Fintype L] [Fintype R] [Fintype C] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| [DecidableEq (p.Vertices L R C)] |
| (lc : LabelCoverData U V L R C) {N : ℕ} |
| (eg : Fin N ≃ GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k)) |
| (u v : p.Vertices L R C) : Decidable ((p.graph lc).Adj u v) := by |
| letI := ShortestPaths.thresholdDecidable (layeredSystem lc p.n) eg |
| (p.locationCode lc u) (gridAntipode (p.locationCode lc v)) |
| exact inferInstanceAs (Decidable (u ≠ v ∧ |
| pathDistance (layeredSystem lc p.n).linkLength (p.locationCode lc u) |
| (gridAntipode (p.locationCode lc v)) ≤ ENNReal.ofReal (1/8))) |
|
|
| def cliqueDecidable (p : SamplerParameters) {U V L R C : Type} |
| [Fintype U] [Fintype V] [Fintype L] [Fintype R] [Fintype C] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| (lc : LabelCoverData U V L R C) {N : ℕ} |
| (eg : Fin N ≃ GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k)) : |
| Decidable (p.CliqueTest lc) := by |
| letI (i : Fin N) := ShortestPaths.thresholdDecidable (layeredSystem lc p.n) eg |
| (eg i) (gridAntipode (eg i)) |
| have he : p.CliqueTest lc ↔ ∃ i : Fin N, |
| pathDistance (layeredSystem lc p.n).linkLength (eg i) (gridAntipode (eg i)) ≤ |
| ENNReal.ofReal (1/8) := by |
| constructor |
| · rintro ⟨x,hx⟩; exact ⟨eg.symm x,by simpa using hx⟩ |
| · rintro ⟨i,hi⟩; exact ⟨eg i,hi⟩ |
| exact decidable_of_iff _ he.symm |
|
|
| def enumeratedGraph (p : SamplerParameters) {U V L R C : Type} |
| [Fintype U] [Fintype V] [Fintype L] [Fintype R] [Fintype C] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| [DecidableEq (p.Vertices L R C)] |
| (lc : LabelCoverData U V L R C) {N t : ℕ} (ht : 0<t) |
| (eg : Fin N ≃ GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k)) |
| (ev : Fin t ≃ p.Vertices L R C) : Graph where |
| vertices := t |
| nonempty := ht |
| adj i j := @decide _ (p.graphAdjDecidable lc eg (ev i) (ev j)) |
| symm i j := by |
| simp only [decide_eq_decide] |
| exact (p.graph lc).adj_comm _ _ |
| loopless i := by simp |
|
|
| def enumeratedOutput (p : SamplerParameters) (q : ℕ) (hq : 0<q) |
| {U V L R C : Type} |
| [Fintype U] [Fintype V] [Fintype L] [Fintype R] [Fintype C] |
| [DecidableEq U] [DecidableEq V] [DecidableEq L] [DecidableEq R] |
| [DecidableEq (p.Vertices L R C)] |
| (lc : LabelCoverData U V L R C) {N t : ℕ} (ht : 0<t) |
| (eg : Fin N ≃ GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k)) |
| (ev : Fin t ≃ p.Vertices L R C) : Graph := by |
| letI := p.cliqueDecidable lc eg |
| exact if p.CliqueTest lc then Graph.clique q hq else p.enumeratedGraph lc ht eg ev |
|
|
| end SamplerParameters |
| end LargeIndependentSets |
|
|
| namespace LargeIndependentSets.Graph |
|
|
| structure Renaming (G H : Graph) where |
| equiv : Fin G.vertices ≃ Fin H.vertices |
| adj_eq : ∀ i j, G.adj i j = H.adj (equiv i) (equiv j) |
|
|
| namespace Renaming |
| variable {G H : Graph} |
|
|
| def symm (e : Renaming G H) : Renaming H G where |
| equiv := e.equiv.symm |
| adj_eq i j := by simpa using (e.adj_eq (e.equiv.symm i) (e.equiv.symm j)).symm |
|
|
| lemma vertices_eq (e : Renaming G H) : G.vertices = H.vertices := by |
| simpa using Fintype.card_congr e.equiv |
|
|
| lemma independent (e : Renaming G H) (A : Finset (Fin G.vertices)) (hA : G.Independent A) : |
| H.Independent (A.image e.equiv) := by |
| intro u hu v hv huv |
| obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hu |
| obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hv |
| rw [← e.adj_eq] |
| exact hA i hi j hj (fun h => huv (congrArg e.equiv h)) |
|
|
| lemma independent_card_le (G : Graph) (A : Finset (Fin G.vertices)) (hA : G.Independent A) : |
| A.card ≤ G.independenceNumber := by |
| unfold Graph.independenceNumber |
| exact Finset.le_sup (f:=Finset.card) (Finset.mem_filter.mpr ⟨by simp,hA⟩) |
|
|
| lemma alpha_le (e : Renaming G H) : G.independenceNumber ≤ H.independenceNumber := by |
| obtain ⟨A,hA,he⟩ := G.exists_max_independent |
| rw [he,← Finset.card_image_of_injective A e.equiv.injective] |
| exact independent_card_le H _ (e.independent A hA) |
|
|
| lemma alpha_eq (e : Renaming G H) : G.independenceNumber = H.independenceNumber := |
| le_antisymm e.alpha_le e.symm.alpha_le |
|
|
| lemma colorable (e : Renaming G H) (h : G.ThreeColorable) : H.ThreeColorable := by |
| obtain ⟨c,hc⟩ := h |
| refine ⟨fun v => c (e.equiv.symm v),?_⟩ |
| intro i j hij |
| apply hc |
| simpa only [e.adj_eq,Equiv.apply_symm_apply] using hij |
|
|
| lemma strict_density {δ : ℝ} (e : Renaming G H) |
| (h : (G.independenceNumber:ℝ) < δ*G.vertices) : |
| (H.independenceNumber:ℝ) < δ*H.vertices := by |
| simpa only [e.alpha_eq,e.vertices_eq] using h |
|
|
| end Renaming |
| end LargeIndependentSets.Graph |
|
|
| end OAI |
|
|
