| import OAI.NumberTheory.Ostmann.Tree.CycleProjection |
| import OAI.NumberTheory.Ostmann.Tree.CycleQuartetActions |
| import OAI.NumberTheory.Ostmann.Tree.QuartetFactorization |
| import OAI.NumberTheory.Ostmann.Tree.TensorActionProjection |
|
|
| namespace OAI |
|
|
| namespace Ostmann.Tree.CycleFiberProjection |
| noncomputable section |
| open scoped BigOperators |
| open QuartetFactorization CycleQuartet |
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable {k b : ℕ} (P : LeafPartition (k+2) b) |
| (c : BalancedSelection P.label (quartetCut k).label) |
|
|
| local instance productFiberFintype {m : Fˣ} : Fintype (ProductFiber m) := Fintype.ofFinite _ |
| local instance cutFiberFintype (totals : Leaves k → Fˣ) : |
| Fintype {M : Leaves (k+2) → Fˣ // (bottomCut k).project M=totals} := Fintype.ofFinite _ |
|
|
| omit [Fintype F] [DecidableEq F] in |
| theorem bottom_project_act (z : Fˣ) (M : Leaves (k+2) → Fˣ) : |
| (bottomCut k).project (c.act z M)=(bottomCut k).project M := by |
| have h := quartet_cycle_project P c z M |
| have he : Density.Cut.project (F := F) (quartetCut k)= |
| Density.Cut.project (F := F) (bottomCut k) := |
| congrArg (fun C : Density.Cut (k+2) k => Density.Cut.project (F := F) C) (cut_unique _ _) |
| simpa only [he] using h |
|
|
| def fiberProjection (totals : Leaves k → Fˣ) (fs : Leaves k → Frame F) |
| (g : F → ℂ) (D : Fˣ) (m : ∀ v, ProductFiber (totals v)) : ℂ := |
| letI : ∀ v, MulAction Fˣ (ProductFiber (totals v)) := fun v => fiberMulAction P c v (totals v) |
| Ostmann.FiniteField.actionProjection (F := F) |
| (fun v (a : ProductFiber (totals v)) => (fs v).value g D a.val) m |
|
|
| theorem projection_zero_of_invalid (T : Diagram F (k+2)) (g : F → ℂ) |
| (totals : Leaves k → Fˣ) |
| (h : propagate k T.parameters T.rootLeft T.rootRight |
| (Density.rootCoefficient T.parameters) totals = none) |
| (M : {M : Leaves (k+2) → Fˣ // (bottomCut k).project M=totals}) : |
| c.projection (T.value g) M.val = 0 := by |
| unfold BalancedSelection.projection Arithmetic.ResidueHaar.average |
| have he (z : Fˣ) : T.value g (c.act z M.val)=0 := by |
| apply diagram_value_zero_of_invalid |
| unfold ancestorFrames |
| rw [bottom_project_act P c,M.property,h] |
| simp only [he,Finset.sum_const_zero,mul_zero] |
|
|
| theorem projection_eq_fiberProjection (T : Diagram F (k+2)) (g : F → ℂ) |
| (totals : Leaves k → Fˣ) (fs : Leaves k → Frame F) |
| (h : propagate k T.parameters T.rootLeft T.rootRight |
| (Density.rootCoefficient T.parameters) totals = some fs) |
| (M : {M : Leaves (k+2) → Fˣ // (bottomCut k).project M=totals}) : |
| c.projection (T.value g) M.val = |
| fiberProjection P c totals fs g T.denominator (productFiberEquiv k totals M) := by |
| unfold BalancedSelection.projection Arithmetic.ResidueHaar.average |
| fiberProjection Ostmann.FiniteField.actionProjection |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro z _ |
| change T.value g (c.act z M.val) = _ |
| rw [diagram_factorization] |
| have he : ancestorFrames T (c.act z M.val)=some fs := by |
| unfold ancestorFrames |
| rw [bottom_project_act P c,M.property,h] |
| rw [he] |
| change (∏ v,(fs v).value g T.denominator (bottomLeaves k (c.act z M.val) v)) = _ |
| apply Finset.prod_congr rfl |
| intro v _ |
| rw [bottomLeaves_act P c] |
| rfl |
|
|
| theorem real_average_equiv {A B : Type*} [Fintype A] [Fintype B] |
| (e : A ≃ B) (f : B → ℝ) : |
| Density.average (fun x => f (e x))=Density.average f := by |
| unfold Density.average |
| rw [Fintype.card_congr e,e.sum_comp] |
|
|
| theorem projection_energy_on_fiber (T : Diagram F (k+2)) (g : F → ℂ) |
| (totals : Leaves k → Fˣ) (fs : Leaves k → Frame F) |
| (h : propagate k T.parameters T.rootLeft T.rootRight |
| (Density.rootCoefficient T.parameters) totals = some fs) : |
| Density.average (fun M : {M : Leaves (k+2) → Fˣ // (bottomCut k).project M=totals} => |
| ‖c.projection (T.value g) M.val‖^2) = |
| Density.average (fun m : ∀ v, ProductFiber (totals v) => |
| ‖fiberProjection P c totals fs g T.denominator m‖^2) := by |
| simp_rw [projection_eq_fiberProjection P c T g totals fs h] |
| have he := real_average_equiv (productFiberEquiv k totals) |
| (fun m => ‖fiberProjection P c totals fs g T.denominator m‖^2) |
| simpa only [real_average_fintype_congr _ |
| (inferInstance : Fintype (∀ v, ProductFiber (totals v)))] using he |
|
|
| theorem projection_energy_zero_of_invalid (T : Diagram F (k+2)) (g : F → ℂ) |
| (totals : Leaves k → Fˣ) |
| (h : propagate k T.parameters T.rootLeft T.rootRight |
| (Density.rootCoefficient T.parameters) totals = none) : |
| Density.average (fun M : {M : Leaves (k+2) → Fˣ // (bottomCut k).project M=totals} => |
| ‖c.projection (T.value g) M.val‖^2) = 0 := by |
| simp only [projection_zero_of_invalid P c T g totals h,norm_zero,zero_pow (by decide : 2 ≠ 0), |
| Density.average,Finset.sum_const_zero,mul_zero] |
|
|
| end |
| end Ostmann.Tree.CycleFiberProjection |
|
|
| end OAI |
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|
