| import OAI.Algebra.Finitistic.Bar.LiteralMultibarCoordinates |
|
|
| namespace OAI |
| noncomputable section |
| namespace DerivedBar.RecursiveTotal |
| open DerivedBar.Multibar |
|
|
| /-- Remove the zero input degree from the list of bar lengths. -/ |
| def moduleBlocks : (r : ℕ) → {n : ℕ} → (t : Index (r+1) n) → Blocks r |
| | 0, _, ⟨p,_⟩ => p.val.1 |
| | r+1, _, ⟨p,t⟩ => (p.val.1, moduleBlocks r t) |
|
|
| theorem total_moduleBlocks (r n : ℕ) (t : Index (r+1) n) : |
| total r (moduleBlocks r t) + inputDegree (r+1) t = n := by |
| induction r generalizing n with |
| | zero => rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩; exact h |
| | succ r ih => |
| rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩ |
| have ht := ih q t |
| change i + total r (moduleBlocks r t) + inputDegree (r+1) t = n |
| omega |
|
|
| theorem pattern_moduleBlocks (r n : ℕ) (t : Index (r+1) n) : |
| pattern r (moduleBlocks r t) = leadingPattern r n t := by |
| induction r generalizing n with |
| | zero => |
| rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩ |
| simp [moduleBlocks, pattern, leadingPattern, innerPattern] |
| | succ r ih => |
| rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩ |
| change List.replicate i false ++ true :: pattern r (moduleBlocks r t) = _ |
| rw [ih q t] |
| rfl |
|
|
| /-- The recursive total index attached to a block tuple, with input degree zero. -/ |
| def moduleIndex : (r : ℕ) → (b : Blocks r) → Index (r+1) (total r b) |
| | 0, i => ⟨⟨⟨i,0⟩,by exact Nat.add_zero i⟩,()⟩ |
| | r+1, (i,b) => ⟨⟨⟨i,total r b⟩,rfl⟩,moduleIndex r b⟩ |
|
|
| @[simp] theorem inputDegree_moduleIndex (r : ℕ) (b : Blocks r) : |
| inputDegree (r+1) (moduleIndex r b) = 0 := by |
| induction r with |
| | zero => rfl |
| | succ r ih => rcases b with ⟨i,b⟩; exact ih b |
|
|
| @[simp] theorem moduleBlocks_moduleIndex (r : ℕ) (b : Blocks r) : |
| moduleBlocks r (moduleIndex r b) = b := by |
| induction r with |
| | zero => rfl |
| | succ r ih => rcases b with ⟨i,b⟩; change (i,moduleBlocks r (moduleIndex r b)) = _; rw [ih] |
|
|
| @[simp] theorem inputDegree_castIndex {r n m : ℕ} (h : n=m) (t : Index r n) : |
| inputDegree r (castIndex h t) = inputDegree r t := by subst m; rfl |
|
|
| @[simp] theorem moduleBlocks_castIndex {r n m : ℕ} (h : n=m) (t : Index (r+1) n) : |
| moduleBlocks r (castIndex h t) = moduleBlocks r t := by subst m; rfl |
|
|
| /-- All indices are recovered from the blocks and input degree; no quotient of |
| block tensors or identification of distinct patterns is introduced. -/ |
| theorem moduleBlocks_injective (r n : ℕ) (t s : Index (r+1) n) |
| (ht : inputDegree (r+1) t=0) (hs : inputDegree (r+1) s=0) |
| (h : moduleBlocks r t=moduleBlocks r s) : t=s := by |
| induction r generalizing n with |
| | zero => |
| rcases t with ⟨⟨⟨i,q⟩,hq⟩,t⟩ |
| rcases s with ⟨⟨⟨j,p⟩,hp⟩,s⟩ |
| change q=0 at ht |
| change p=0 at hs |
| subst q p |
| have hij : i=j := by omega |
| subst j |
| cases t; cases s; rfl |
| | succ r ih => |
| rcases t with ⟨⟨⟨i,q⟩,hq⟩,t⟩ |
| rcases s with ⟨⟨⟨j,p⟩,hp⟩,s⟩ |
| change (i,moduleBlocks r t) = (j,moduleBlocks r s) at h |
| have hij := congrArg Prod.fst h |
| dsimp at hij |
| subst j |
| have hqp : q=p := by omega |
| subst p |
| have hts := ih q t s ht hs (congrArg Prod.snd h) |
| subst s |
| rfl |
|
|
| /-- Exact module-input indexing by the manuscript's tuples of r+1 block lengths. -/ |
| def moduleBlockEquiv (r n : ℕ) : |
| {t : Index (r+1) n // inputDegree (r+1) t=0} ≃ |
| {b : Blocks r // total r b=n} where |
| toFun t := ⟨moduleBlocks r t.val, by |
| have h := total_moduleBlocks r n t.val |
| rw [t.property, Nat.add_zero] at h |
| exact h⟩ |
| invFun b := ⟨castIndex b.property (moduleIndex r b.val), by simp⟩ |
| left_inv t := by |
| apply Subtype.ext |
| apply moduleBlocks_injective r n _ _ (by simp) t.property |
| simp |
| right_inv b := by apply Subtype.ext; simp |
|
|
| /-- Exact full-degree shape index, including the shift by the number of X slots. -/ |
| def moduleShapeEquiv (r n : ℕ) : |
| {t : Index (r+1) n // inputDegree (r+1) t=0} ≃ |
| Multibar.Indexed.Shape (r+n) r := |
| (moduleBlockEquiv r n).trans (Multibar.Indexed.blockDegreeEquiv r n) |
| end DerivedBar.RecursiveTotal |
|
|
| end |
|
|
| end OAI |
|
|
