| import OAI.NumberTheory.Ostmann.Tree.DensityCut |
|
|
| namespace OAI |
|
|
| namespace Ostmann.Tree.Density.Cut |
| open scoped BigOperators |
|
|
| def label : {d k : ℕ} → Cut d k → Leaves d → Leaves k |
| | _,_,.stop _,_ => fun i => Fin.elim0 i |
| | _,_,.split L R,v => if v 0 then Fin.cons true (label R (Fin.tail v)) |
| else Fin.cons false (label L (Fin.tail v)) |
|
|
| @[simp] theorem label_cons_false {d k : ℕ} (L R : Cut d k) (v : Leaves d) : |
| label (.split L R) (Fin.cons false v)=Fin.cons false (label L v) := by simp [label] |
| @[simp] theorem label_cons_true {d k : ℕ} (L R : Cut d k) (v : Leaves d) : |
| label (.split L R) (Fin.cons true v)=Fin.cons true (label R v) := by simp [label] |
|
|
| theorem label_surjective {d k : ℕ} (C : Cut d k) : Function.Surjective (label C) := by |
| induction C with |
| | stop d => |
| intro v |
| exact ⟨fun _ => false,Subsingleton.elim _ _⟩ |
| | @split d k L R ihl ihr => |
| intro v |
| obtain ⟨⟨b,w⟩,rfl⟩ := (Density.pathEquiv k).surjective v |
| cases b with |
| | false => |
| obtain ⟨e,he⟩ := ihl w |
| refine ⟨Fin.cons false e, ?_⟩ |
| change label (.split L R) (Fin.cons false e)=Fin.cons false w |
| rw [label_cons_false,he] |
| | true => |
| obtain ⟨e,he⟩ := ihr w |
| refine ⟨Fin.cons true e, ?_⟩ |
| change label (.split L R) (Fin.cons true e)=Fin.cons true w |
| rw [label_cons_true,he] |
|
|
| theorem project_eq_label_product {F : Type*} [Field F] {d k : ℕ} (C : Cut d k) |
| (M : Leaves d → Fˣ) (v : Leaves k) : |
| project C M v = ∏ e with label C e=v, M e := by |
| classical |
| induction C with |
| | stop d => |
| have hv : ∀e : Leaves d,label (.stop d) e=v := fun _ => Subsingleton.elim _ _ |
| simp [project,Parameters.leafProduct,hv] |
| | @split d k L R ihl ihr => |
| obtain ⟨⟨b,w⟩,rfl⟩ := (Density.pathEquiv k).surjective v |
| change project (.split L R) M (Fin.cons b w) = |
| ∏ e with label (.split L R) e=Fin.cons b w, M e |
| cases b <;> |
| simp only [project,Density.join,Fin.cons_zero,Fin.tail_cons, |
| Bool.false_eq_true,↓reduceIte] <;> |
| rw [Finset.prod_filter,Density.prod_split] <;> |
| simp only [Density.left,Density.right,label_cons_false,label_cons_true, |
| Fin.cons_inj, Bool.false_eq_true, Bool.true_eq_false, false_and, true_and, |
| ↓reduceIte,Finset.prod_const_one,mul_one,one_mul] <;> |
| rw [← Finset.prod_filter] |
| · exact ihl _ _ |
| · exact ihr _ _ |
|
|
| end Ostmann.Tree.Density.Cut |
|
|
| end OAI |
|
|
