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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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
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