| import OAI.Algebra.Finitistic.Ordinary.OrdinaryChainAlgebra |
| import OAI.Algebra.Finitistic.Encoding.DirectedBound |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.OrdinarySimulation.ChainAlgebra |
| universe u |
| variable (k : Type u) [Field k] (l : ℕ) |
|
|
| theorem vertex_orthogonal (i j : Fin (l+1)) (hij : i ≠ j) : vertex k l i * vertex k l j = 0 := by |
| apply Prod.ext |
| · funext t |
| change (Pi.single i (1:k) : Fin (l+1) → k) t * (Pi.single j (1:k) : Fin (l+1) → k) t = 0 |
| by_cases ht : t=i |
| · subst t; simp [hij.symm] |
| · simp [ht] |
| · funext t |
| change (Pi.single i (1:k) : Fin (l+1) → k) t.succ * 0 + 0 * (Pi.single j (1:k) : Fin (l+1) → k) t.castSucc = 0 |
| simp |
|
|
| theorem vertex_complete : ∑ i : Fin (l+1), vertex k l i = 1 := by |
| apply Prod.ext |
| · funext t |
| have h := map_sum ((LinearMap.proj t).comp |
| (LinearMap.fst k (Fin (l+1) → k) (Fin l → k))) (fun i : Fin (l+1) => vertex k l i) Finset.univ |
| change (∑ i : Fin (l+1), vertex k l i).1 t = 1 |
| exact h.trans (by |
| change (∑ i : Fin (l+1), (Pi.single i (1:k) : Fin (l+1) → k) t) = 1 |
| simp [Pi.single_apply]) |
| · funext t |
| have h := map_sum ((LinearMap.proj t).comp |
| (LinearMap.snd k (Fin (l+1) → k) (Fin l → k))) (fun i : Fin (l+1) => vertex k l i) Finset.univ |
| change (∑ i : Fin (l+1), vertex k l i).2 t = 0 |
| exact h.trans (by |
| change (∑ _i : Fin (l+1), (0:k)) = 0 |
| simp) |
|
|
| theorem vertex_corner_diagonal (i : Fin (l+1)) (x : ChainAlgebra k l) : |
| vertex k l i * x * vertex k l i = x.1 i • vertex k l i := by |
| apply Prod.ext |
| · funext t |
| change ((Pi.single i (1:k) : Fin (l+1) → k) t * x.1 t) * |
| (Pi.single i (1:k) : Fin (l+1) → k) t = x.1 i * (Pi.single i (1:k) : Fin (l+1) → k) t |
| by_cases ht : t=i <;> simp [ht] |
| · funext t |
| change ((Pi.single i (1:k) : Fin (l+1) → k) t.succ * x.1 t.succ) * 0 + |
| ((Pi.single i (1:k) : Fin (l+1) → k) t.succ * x.2 t + 0 * x.1 t.castSucc) * |
| (Pi.single i (1:k) : Fin (l+1) → k) t.castSucc = x.1 i * 0 |
| by_cases h₁ : t.succ = i |
| · have h₂ : t.castSucc ≠ i := by intro h; have := congrArg Fin.val h₁; have := congrArg Fin.val h; simp at *; omega |
| simp [h₂] |
| · simp [h₁] |
|
|
| theorem vertex_directed (i j : Fin (l+1)) (hij : j < i) (x : ChainAlgebra k l) : |
| vertex k l j * x * vertex k l i = 0 := by |
| apply Prod.ext |
| · funext t |
| change ((Pi.single j (1:k) : Fin (l+1) → k) t * x.1 t) * |
| (Pi.single i (1:k) : Fin (l+1) → k) t = 0 |
| by_cases ht : t=j |
| · subst t; simp [ne_of_lt hij] |
| · simp [ht] |
| · funext t |
| change ((Pi.single j (1:k) : Fin (l+1) → k) t.succ * x.1 t.succ) * 0 + |
| ((Pi.single j (1:k) : Fin (l+1) → k) t.succ * x.2 t + 0 * x.1 t.castSucc) * |
| (Pi.single i (1:k) : Fin (l+1) → k) t.castSucc = 0 |
| by_cases h₁ : t.succ=j |
| · have h₂ : t.castSucc ≠ i := by |
| intro h |
| have := congrArg Fin.val h₁ |
| have := congrArg Fin.val h |
| change j.val < i.val at hij |
| simp at * |
| omega |
| simp [h₂] |
| · simp [h₁] |
|
|
| / |
| def directedSystem : DirectedBound.DirectedSystem k (ChainAlgebra k l) (l+1) where |
| e := vertex k l |
| idem := vertex_idempotent k l |
| orthogonal := vertex_orthogonal k l |
| complete := vertex_complete k l |
| diagonal i x := ⟨x.1 i,vertex_corner_diagonal k l i x⟩ |
| directed := vertex_directed k l |
| end LittleFinitistic.OrdinarySimulation.ChainAlgebra |
|
|
| end |
|
|
| end OAI |
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|
