| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Geometry.EmbeddingLayerLaw |
| import OAI.Combinatorics.IndependentSets.PCP.CircleRound |
| import OAI.Combinatorics.IndependentSets.Reduction.SwapOccurrence |
| import OAI.Combinatorics.IndependentSets.Expansion.Supported2 |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| open scoped Classical BigOperators |
| open MeasureTheory PhaseTest |
|
|
| structure SamplerParameters where |
| n : ℕ |
| s : ℕ |
| hs : 0 < s |
| m : ℕ |
| hm : 0 < m |
| k : ℕ |
| hk : 1 ≤ k |
| ρ : ℚ |
| hρ : 0 ≤ ρ |
| μ : LayerLaw (n+1) s |
|
|
| namespace SamplerParameters |
| instance (p : SamplerParameters) : NeZero (p.m*2^p.k) := |
| ⟨Nat.ne_of_gt (Nat.mul_pos p.hm (by positivity))⟩ |
| |
| lemma even (p : SamplerParameters) : Even (p.m*2^p.k) := by |
| obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt (lt_of_lt_of_le Nat.zero_lt_one p.hk)) |
| refine ⟨p.m*2^k, ?_⟩ |
| rw [hk,pow_succ] |
| ring |
| |
| noncomputable abbrev law (p : SamplerParameters) (L R : Type) [Fintype L] [Fintype R] := |
| samplerLaw (L:=L) (R:=R) p.μ p.m p.k p.hρ |
| |
| abbrev Vertices (p : SamplerParameters) (L R C : Type) [Fintype L] [Fintype R] := |
| (p.law L R).Vertices C p.n |
| |
| noncomputable def location (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) := |
| sampleLocation lc v.1 p.hs v.2.1 |
| |
| noncomputable def graph (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) : SimpleGraph (p.Vertices L R C) := |
| gridGraph (layeredSystem lc p.n) p.even (p.location lc) |
| |
| theorem completeness (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (ℓ : U → L) (ρ : V → R) (h : lc.Satisfied ℓ ρ) : |
| (∀ x : GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k), |
| ¬pathDistance (layeredSystem lc p.n).linkLength x (gridAntipode x) ≤ ENNReal.ofReal (1/8)) ∧ |
| ∃ color : p.Vertices L R C → Fin 3, ∀ u v, (p.graph lc).Adj u v → color u ≠ color v := by |
| have hC := layered_completeness lc p.n ℓ ρ h p.even |
| exact ⟨hC.1, hC.2 _ (p.location lc)⟩ |
| |
| noncomputable def functions (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (A : Set (p.Vertices L R C)) |
| (q : LayerQuestion U V p.n) : (LayerAnswer L R q → Circle) → ℝ := |
| tupleExtension (layeredSystem lc p.n) (p.location lc '' A) q |
| |
| lemma functions_lipschitz (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (A : Set (p.Vertices L R C)) (q : LayerQuestion U V p.n) : |
| LipschitzWith 64 (p.functions lc A q) := |
| tupleExtension_lipschitz _ _ _ |
| |
| lemma functions_bounded (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (A : Set (p.Vertices L R C)) |
| (q : LayerQuestion U V p.n) (x : LayerAnswer L R q → Circle) : |
| |p.functions lc A q x| ≤ 1 := |
| abs_le.mpr (tupleExtension_bounds _ _ _ _) |
| |
| lemma functions_odd (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) (A : Set (p.Vertices L R C)) |
| (q : LayerQuestion U V p.n) (x : LayerAnswer L R q → Circle) : |
| p.functions lc A q (fun z => x z + ((1/2:ℝ):Circle)) = -p.functions lc A q x := |
| tupleExtension_odd _ _ _ _ |
| |
| lemma functions_compatibility (p : SamplerParameters) {U V L R C : Type} |
| [Fintype L] [Fintype R] (lc : LabelCoverData U V L R C) |
| (hnoclique : ∀ x : GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k), |
| ¬pathDistance (layeredSystem lc p.n).linkLength x (gridAntipode x) ≤ ENNReal.ofReal (1/8)) |
| (A : Set (p.Vertices L R C)) (hA : (p.graph lc).IsIndepSet A) |
| (chain : Fin p.n → C) (i j : Fin (p.n+1)) (hij : i ≤ j) |
| (y : MixedTuple L R p.n j.val → Circle) : |
| |p.functions lc A (occurrenceQuestion lc chain i) (y ∘ chainComposite lc chain i.val j.val hij) - |
| p.functions lc A (occurrenceQuestion lc chain j) y| ≤ 2*64/(p.m*2^p.k:ℕ) := by |
| rw [abs_sub_comm] |
| have hpath (v : MixedTuple L R p.n j.val → ZMod (p.m*2^p.k)) : |
| pathDistance (layeredSystem lc p.n).linkLength |
| ⟨occurrenceQuestion lc chain i, v ∘ chainComposite lc chain i.val j.val hij⟩ |
| ⟨occurrenceQuestion lc chain j, v⟩ = 0 := |
| occurrence_chain_path_zero lc chain i.val j.val hij (Nat.le_of_lt_succ j.isLt) v |
| have hh := tupleExtension_compatibility (layeredSystem lc p.n) (D:=p.m*2^p.k) |
| p.even hnoclique (p.location lc) A hA |
| (q:=occurrenceQuestion lc chain i) (q':=occurrenceQuestion lc chain j) |
| (chainComposite lc chain i.val j.val hij) hpath y |
| exact hh |
| |
| theorem density_le_fibers (p : SamplerParameters) {U V L R C : Type} |
| [Fintype L] [Fintype R] [Fintype C] (lc : LabelCoverData U V L R C) |
| (hnoclique : ∀ x : GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k), |
| ¬pathDistance (layeredSystem lc p.n).linkLength x (gridAntipode x) ≤ ENNReal.ofReal (1/8)) |
| (A : Finset (p.Vertices L R C)) (hA : (p.graph lc).IsIndepSet (A : Set _)) : |
| (A.card : ℝ) / Fintype.card (p.Vertices L R C) ≤ |
| 𝔼 chain : Fin p.n → C, ∑ B, (p.μ.supported.weight B : ℝ) * |
| fiberSquare lc chain (p.functions lc A) B.val (supported_nonempty p.hs B) p.m p.ρ p.k := by |
| let F := p.functions lc (A : Set _) |
| have hon (v : p.Vertices L R C) (hv : v ∈ A) : |
| F (p.location lc v).1 (torusGrid (p.location lc v).2) = 1 := by |
| rw [show F = tupleExtension (layeredSystem lc p.n) (p.location lc '' (A : Set _)) from rfl] |
| rw [tupleExtension_on_grid (layeredSystem lc p.n) p.even hnoclique (p.location lc) A hA] |
| exact (grid_independent_bump (layeredSystem lc p.n) p.even hnoclique (p.location lc) A hA).2.2.1 v hv |
| have hden := density_le_location_square A (p.location lc) |
| (fun x => F x.1 (torusGrid x.2)) hon |
| apply hden.trans |
| change (𝔼 v : (p.law L R).Vertices C p.n, |
| (F (sampleLocation lc v.1 p.hs v.2.1).1 |
| (torusGrid (sampleLocation lc v.1 p.hs v.2.1).2))^2) ≤ _ |
| rw [(p.law L R).expect_vertices p.n (fun chain o => |
| (F (sampleLocation lc chain p.hs o).1 (torusGrid (sampleLocation lc chain p.hs o).2))^2)] |
| apply le_of_eq |
| apply Finset.expect_congr rfl |
| intro chain _ |
| exact sampler_expectation lc chain p.μ p.hρ p.hs p.hm F |
| |
| end SamplerParameters |
| |
| theorem finite_source_soundness {δ : ℝ} (hδ : 0 < δ) : |
| ∃ p : SamplerParameters, ∃ σ : ℚ, 0 < σ ∧ σ < 1 ∧ |
| ∀ (U V L R C : Type) [Fintype L] [Fintype R] [Fintype C] |
| [Nonempty L] [Nonempty R] [Nonempty C] (lc : LabelCoverData U V L R C), |
| lc.Sound (σ:ℝ) → |
| (∀ x : GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k), |
| ¬pathDistance (layeredSystem lc p.n).linkLength x (gridAntipode x) ≤ ENNReal.ofReal (1/8)) → |
| ∀ A : Finset (p.Vertices L R C), (p.graph lc).IsIndepSet (A : Set _) → |
| (A.card : ℝ) / Fintype.card (p.Vertices L R C) < δ := by |
| let ε := δ/10 |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| obtain ⟨β,hβ,K,hK,HT⟩ := actual_aligned_fiber 64 hε |
| obtain ⟨m,hm,ρ,hρ,hρ1,hstep,hρK,s,hs,hrare,γ,hγ,hzero,k,hk,hD,hgrid⟩ := |
| finite_parameter_selection (L:=64) hε hβ (by norm_num) K |
| obtain ⟨J,hJ,HJ⟩ := canonical_subchain_rules 64 s hγ |
| let d := max 1 (J*(m+1)^s) |
| have hd : 1 ≤ d := le_max_left _ _ |
| have hs0 : 0 < s := by omega |
| obtain ⟨r,hr,μ,hμ⟩ := distributional_alignment s d (by omega) hd ε hε |
| cases r with |
| | zero => omega |
| | succ n => |
| obtain ⟨σ,hσ,hσ1,hσb⟩ := final_soundness_choice hε (n+1) d |
| let p : SamplerParameters := ⟨n,s,hs0,m,hm,k,hk,ρ,hρ.le,μ⟩ |
| refine ⟨p,σ,hσ,hσ1,?_⟩ |
| intro U V L R C _ _ _ _ _ _ lc hsound hnoclique A hA |
| let F := p.functions lc (A : Set _) |
| have hF : ∀ q, LipschitzWith 64 (F q) := p.functions_lipschitz lc A |
| have hb : ∀ q x, |F q x| ≤ 1 := p.functions_bounded lc A |
| have hodd : ∀ q x, F q (fun z => x z + ((1/2:ℝ):Circle)) = -F q x := |
| p.functions_odd lc A |
| obtain ⟨rule,Hrule⟩ := HJ U V L R n F hF hb m hm |
| let rules := ruleUnion rule |
| let lists (chain : Fin n → C) (I : Finset (Fin (n+1))) := |
| terminalList lc chain I (rules I (chainView lc chain I)) |
| have hrcard (I : Finset (Fin (n+1))) (hI : Small s I) (v : SubchainView U V L R n I) : |
| (rules I v).card ≤ d := by |
| change (ruleUnion rule I v).card ≤ d |
| simp only [ruleUnion, dite_eq_left hI] |
| exact (Hrule I hI v).2.trans (le_max_right _ _) |
| have hlcard (chain : Fin n → C) (I : Finset (Fin (n+1))) (hI : Small s I) : |
| (lists chain I).card ≤ d := |
| Finset.card_image_le.trans (hrcard I hI (chainView lc chain I)) |
| have hbad : (𝔼 chain : Fin n → C, (μ.badMass (lists chain) : ℝ)) ≤ |
| ε + 4^(n+1)*(d:ℝ)^2*σ := by |
| have hm' : ∀ W : Finset (Fin (n+1)) → Finset (MixedTuple L R n n), |
| (∀ I, Small s I → (W I).card ≤ d) → ListsSeparated (s:=s) W → (μ.badMass W : ℝ) ≤ ε := by |
| intro W hc hsep |
| exact hμ (MixedTuple L R n n) W hc hsep |
| exact good_pair_probability μ hε.le (by exact_mod_cast hσ.le) hm' lists hlcard |
| (fun I J hI hJ hsep => source_lists_probability lc hsound rules hrcard I J hI hJ hsep) |
| have hbad2 : (𝔼 chain : Fin n → C, ∑ B, |
| (μ.supported.weight B : ℝ) * (if ¬Aligned (lists chain) B.val then 1 else 0)) ≤ 2*ε := by |
| have hbeq : (𝔼 chain : Fin n → C, ∑ B, |
| (μ.supported.weight B : ℝ) * (if ¬Aligned (lists chain) B.val then 1 else 0)) = |
| 𝔼 chain : Fin n → C, (μ.badMass (lists chain) : ℝ) := by |
| apply Finset.expect_congr rfl |
| intro chain _ |
| exact μ.bad_supported (lists chain) |
| rw [hbeq] |
| linarith |
| have hgood (chain : Fin n → C) (B : {B : Finset (Fin (n+1)) // B.card = s}) |
| (h : ¬¬Aligned (lists chain) B.val) : |
| fiberSquare lc chain F B.val (supported_nonempty hs0 B) m ρ k ≤ 6*ε := by |
| have herr : (0:ℝ) ≤ 2*64/(m*2^k:ℕ) := by positivity |
| have herrγ : (2*64:ℝ)/(m*2^k:ℕ) < γ := by exact_mod_cast hD |
| have hkgrid : (2*64:ℝ)*s*(1/2:ℝ)^k < ε := by |
| convert hgrid using 1 |
| simp |
| ring |
| exact (HT m s k ρ γ (2*64/(m*2^k:ℕ)) hm hs (by exact_mod_cast hρ.le) |
| herr herrγ hstep hrare hρK hzero hkgrid U V L R C n lc chain F hF hb hodd |
| (fun i j hij y => p.functions_compatibility lc hnoclique A hA chain i j hij y) |
| rule (fun I hI v t => ((Hrule I hI v).1 t).2) |
| B.val (supported_nonempty hs0 B) B.property (Classical.not_not.mp h)).le |
| apply density_from_good_fibers μ.supported |
| (fun chain B => ¬Aligned (lists chain) B.val) |
| (fun chain B => fiberSquare lc chain F B.val (supported_nonempty hs0 B) m ρ k) |
| hε.le (by dsimp [ε]; linarith) (by |
| calc |
| _ = (𝔼 chain : Fin n → C, ∑ B, (μ.supported.weight B : ℝ) * |
| (if ¬Aligned (lists chain) B.val then 1 else 0)) := by |
| apply Finset.expect_congr rfl |
| intro chain _ |
| apply Finset.sum_congr rfl |
| intro B _ |
| by_cases h : Aligned (lists chain) B.val <;> simp [h] |
| _ ≤ 2*ε := hbad2) hgood |
| (fun chain B => fiberSquare_le_one lc chain F hb B.val (supported_nonempty hs0 B) m k |
| (by exact_mod_cast hρ.le)) A |
| exact p.density_le_fibers lc hnoclique A hA |
| |
| end LargeIndependentSets |
| |
| end OAI |
| |
