| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import OAI.Computability.UniqueGames.Inverse.KMSFoundationLemmas |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| |
|
|
| Distinct lines always intersect trivially, so this Grassmann graph is complete. |
| Its retention never exceeds global density. This proves a genuine restricted |
| expansion statement; it does not imply the OPEN large-dimension KMS principle. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Inverse.KMS |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| theorem adjacent_one_iff {n : ℕ} (L M : Vertex n 1) : |
| Adjacent L M ↔ L ≠ M := by |
| constructor |
| · exact fun h => h.1 |
| · intro hne |
| refine ⟨hne, ?_⟩ |
| have hle : Module.finrank F2 ↥(L.val ⊓ M.val : Submodule F2 (Ambient n)) ≤ 1 := |
| (Submodule.finrank_mono inf_le_left).trans_eq L.property |
| have hnot : Module.finrank F2 ↥(L.val ⊓ M.val : Submodule F2 (Ambient n)) ≠ 1 := by |
| intro heq |
| have hL : L.val ⊓ M.val = L.val := |
| Submodule.eq_of_le_of_finrank_eq inf_le_left (heq.trans L.property.symm) |
| have hM : L.val ⊓ M.val = M.val := |
| Submodule.eq_of_le_of_finrank_eq inf_le_right (heq.trans M.property.symm) |
| exact hne (Subtype.ext (hL.symm.trans hM)) |
| omega |
|
|
| theorem neighbors_one_eq {n : ℕ} (L : Vertex n 1) : |
| neighbors L = Finset.univ.erase L := by |
| ext M |
| simp [neighbors, adjacent_one_iff, ne_comm] |
|
|
| private theorem inter_univ_erase {Ω : Type*} [Fintype Ω] [DecidableEq Ω] |
| (S : Finset Ω) (x : Ω) : S ∩ Finset.univ.erase x = S.erase x := by |
| ext y |
| simp [and_comm] |
|
|
| theorem dimensionOne_neighbor_density {n : ℕ} (S : Finset (Vertex n 1)) |
| {L : Vertex n 1} (hL : L ∈ S) : |
| relativeDensity S (neighbors L) = |
| ((S.card : ℝ) - 1) / ((Fintype.card (Vertex n 1) : ℝ) - 1) := by |
| unfold relativeDensity |
| rw [neighbors_one_eq, inter_univ_erase, Finset.card_erase_of_mem hL, |
| Finset.card_erase_of_mem (Finset.mem_univ L), Finset.card_univ] |
| have hS : 1 ≤ S.card := Finset.one_le_card.mpr ⟨L, hL⟩ |
| have hV : 1 ≤ Fintype.card (Vertex n 1) := Fintype.card_pos_iff.mpr ⟨L⟩ |
| rw [Nat.cast_sub hS, Nat.cast_sub hV] |
| norm_num |
|
|
| /-- Exact retention of a nonempty collection of binary lines. -/ |
| theorem dimensionOne_retention {n : ℕ} (S : Finset (Vertex n 1)) |
| (hS : S.Nonempty) : |
| retention S = ((S.card : ℝ) - 1) / |
| ((Fintype.card (Vertex n 1) : ℝ) - 1) := by |
| have hsum : (∑ L ∈ S, relativeDensity S (neighbors L)) = |
| (S.card : ℝ) * (((S.card : ℝ) - 1) / |
| ((Fintype.card (Vertex n 1) : ℝ) - 1)) := by |
| calc |
| _ = ∑ _L ∈ S, ((S.card : ℝ) - 1) / |
| ((Fintype.card (Vertex n 1) : ℝ) - 1) := |
| Finset.sum_congr rfl fun _ hL => dimensionOne_neighbor_density S hL |
| _ = _ := by simp |
| unfold retention |
| rw [hsum] |
| exact mul_div_cancel_left₀ _ (Nat.cast_ne_zero.mpr (Finset.card_ne_zero.mpr hS)) |
|
|
| /-- The one-dimensional Grassmann graph has retention at most global density. |
| The cardinality hypothesis rules out the single-vertex, zero-degree case. -/ |
| theorem dimensionOne_retention_le_density {n : ℕ} (S : Finset (Vertex n 1)) |
| (hS : S.Nonempty) (hV : 1 < Fintype.card (Vertex n 1)) : |
| retention S ≤ relativeDensity S Finset.univ := by |
| rw [dimensionOne_retention S hS] |
| simp only [relativeDensity, Finset.inter_univ, Finset.card_univ] |
| have hV' : (1 : ℝ) < Fintype.card (Vertex n 1) := Nat.one_lt_cast.mpr hV |
| have hcard : (S.card : ℝ) ≤ Fintype.card (Vertex n 1) := |
| Nat.cast_le.mpr (Finset.card_le_univ S) |
| apply (div_le_div_iff₀ (by linarith) (by linarith)).mpr |
| nlinarith |
|
|
| /-- In dimension one, the whole ambient interval already has the required |
| density. This is a proved special case, not a premise for higher dimensions. -/ |
| theorem dimensionOne_dense_interval {n : ℕ} (S : Finset (Vertex n 1)) |
| (hS : S.Nonempty) (hV : 1 < Fintype.card (Vertex n 1)) |
| {ζ : ℝ} (hζ : ζ ≤ retention S) : |
| ζ ≤ relativeDensity S (interval (⊥ : Submodule F2 (Ambient n)) ⊤) := by |
| have hI : interval (ell := 1) (⊥ : Submodule F2 (Ambient n)) ⊤ = |
| Finset.univ := by simp [interval] |
| rw [hI] |
| exact hζ.trans (dimensionOne_retention_le_density S hS hV) |
|
|
| end |
| end UniqueGamesTheorem.Inverse.KMS |
|
|
| end |
|
|
| end OAI |
|
|
