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216fb31 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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]
/-- The actual chain algebra has the source's directed vertex system, including l=0. -/
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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