| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas |
| import Mathlib.LinearAlgebra.Dimension.Free |
| import Mathlib.LinearAlgebra.Basis.VectorSpace |
| import Mathlib.Algebra.BigOperators.Group.Finset.Pi |
| import Mathlib.Basic.Real.Basic |
| import Lean.Elab.Tactic.Omega |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz |
| open Module Submodule |
| open scoped BigOperators |
| variable {W : Type*} [AddCommGroup W] [Module ℝ W] [FiniteDimensional ℝ W] |
|
|
| theorem exists_adapted_family (E : ℕ → Submodule ℝ W) (hE : Monotone E) (hzero : E 0 = ⊥) |
| (h : ℕ) : |
| ∃ (v : Fin (finrank ℝ (E h)) → W) (d : Fin (finrank ℝ (E h)) → ℕ), |
| LinearIndependent ℝ v ∧ span ℝ (Set.range v) = E h ∧ |
| (∀ i, d i < h ∧ v i ∈ E (d i+1)) ∧ |
| (∑ i, d i) = ∑ e ∈ Finset.range h, e * (finrank ℝ (E (e+1)) - finrank ℝ (E e)) := by |
| classical |
| induction h with |
| | zero => |
| have hd : finrank ℝ (E 0) = 0 := by simp [hzero] |
| rw [hd] |
| refine ⟨Fin.elim0, Fin.elim0, linearIndependent_empty_type, ?_, ?_, ?_⟩ |
| · simp [hzero] |
| · intro i |
| exact Fin.elim0 i |
| · simp |
| | succ h ih => |
| obtain ⟨v, d, hli, hspan, hmem, hsum⟩ := ih |
| obtain ⟨C, hdisj, hsup⟩ := IsModularLattice.exists_disjoint_and_sup_eq (hE (Nat.le_succ h)) |
| have hCle : C ≤ E (h+1) := by rw [← hsup]; exact le_sup_right |
| have hdim : finrank ℝ (E h) + finrank ℝ C = finrank ℝ (E (h+1)) := by |
| have hh := Submodule.finrank_sup_add_finrank_inf_eq (E h) C |
| rw [hsup, hdisj.eq_bot, finrank_bot, add_zero] at hh |
| exact hh.symm |
| have hCdim : finrank ℝ C = finrank ℝ (E (h+1)) - finrank ℝ (E h) := by omega |
| let b := finBasis ℝ C |
| let w : Fin (finrank ℝ C) → W := fun i => C.subtype (b i) |
| have hwli : LinearIndependent ℝ w := b.linearIndependent.map' C.subtype (Submodule.ker_subtype C) |
| have hwspan : span ℝ (Set.range w) = C := by |
| change span ℝ (Set.range (C.subtype ∘ b)) = C |
| rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top] |
| let v' := Sum.elim v w |
| let d' := Sum.elim d (fun _ : Fin (finrank ℝ C) => h) |
| have hv' : LinearIndependent ℝ v' := |
| hli.sum_type hwli (by rwa [hspan, hwspan]) |
| let e : Fin (finrank ℝ (E (h+1))) ≃ |
| (Fin (finrank ℝ (E h)) ⊕ Fin (finrank ℝ C)) := |
| Fintype.equivOfCardEq (by simp [hdim]) |
| refine ⟨v' ∘ e, d' ∘ e, hv'.comp e e.injective, ?_, ?_, ?_⟩ |
| · rw [Set.range_comp, e.surjective.range_eq, Set.image_univ] |
| change span ℝ (Set.range (Sum.elim v w)) = E (h+1) |
| rw [Set.Sum.elim_range, span_union, hspan, hwspan, hsup] |
| · intro i |
| change d' (e i) < h+1 ∧ v' (e i) ∈ E (d' (e i)+1) |
| cases he : e i with |
| | inl j => |
| change d j < h+1 ∧ v j ∈ E (d j+1) |
| exact ⟨(hmem j).1.trans (Nat.lt_succ_self h), (hmem j).2⟩ |
| | inr j => |
| change h < h+1 ∧ w j ∈ E (h+1) |
| exact ⟨Nat.lt_succ_self h, hCle (b j).property⟩ |
| · change (∑ i, d' (e i)) = _ |
| rw [e.sum_comp, Fintype.sum_sum_type, Finset.sum_range_succ] |
| change (∑ i, d i) + (∑ _ : Fin (finrank ℝ C), h) = _ |
| rw [hsum] |
| simp [hCdim, Nat.mul_comm] |
|
|
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
|
|
