File size: 4,880 Bytes
a72760f | 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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
|