File size: 4,127 Bytes
9b6e014 | 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 99 100 101 102 103 104 105 106 107 | import Mathlib
import OAI.Probability.BinarySweep.Mixing.BinaryLineDisk
import OAI.Probability.BinarySweep.FiniteLaws.IndependentOperator
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
variable {b : ℕ} (bits : Fin b → ℕ)
def gridStageHom (j : Fin b) :
(GridOutside bits j → Equiv.Perm (Slot (bits j))) →* Equiv.Perm (GridSlot bits) where
toFun f := (Equiv.piSplitAt j (fun i => Slot (bits i))).symm.permCongr
(Equiv.prodCongrLeft f)
map_one' := by
have h : Equiv.prodCongrLeft (1 : GridOutside bits j → Equiv.Perm (Slot (bits j)))=1 := by
ext x <;> rfl
change (Equiv.piSplitAt j (fun i => Slot (bits i))).symm.permCongrHom
(Equiv.prodCongrLeft 1)=1
rw [h,map_one]
map_mul' f g := by
ext x
simp [Equiv.permCongr_apply,Equiv.prodCongrLeft_apply]
lemma gridStageHom_layer (g : GridChoices bits) (j : Fin b) :
gridStageHom bits j (g j)=gridLayer bits g j := rfl
def complexGridWeight (z : ℂ) (g : GridChoices bits) : ℂ :=
∏j, ∏y, complexLineLaw (bits j) z (g j y)
lemma complexGridWeight_real (z : ℝ) (g : GridChoices bits) :
complexGridWeight bits (z:ℂ) g=(gridWeight bits z g:ℂ) := by
simp [complexGridWeight,gridWeight,complexLineLaw_real]
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) V)
def complexGridAverage (z : ℂ) : V →L[ℂ] V :=
∑g : GridChoices bits, complexGridWeight bits z g • continuousRep ρ (gridSweep bits g)
lemma complexGridAverage_product (z : ℂ) :
complexGridAverage bits ρ z=(List.ofFn (fun j =>
complexAverage (ρ.comp (gridStageHom bits j))
(fun f => ∏y,complexLineLaw (bits j) z (f y)))).reverse.prod := by
unfold complexGridAverage complexGridWeight gridSweep
simp_rw [map_list_prod,List.map_reverse,List.map_ofFn,Function.comp_def]
change (∑g : GridChoices bits, (∏j, ∏y, complexLineLaw (bits j) z (g j y)) •
(List.ofFn (fun j => continuousRep ρ (gridStageHom bits j (g j)))).reverse.prod)=_
rw [independent_average_product (X := fun j => GridOutside bits j → Equiv.Perm (Slot (bits j)))
(fun j f => ∏y,complexLineLaw (bits j) z (f y))
(fun j f => continuousRep ρ (gridStageHom bits j f))]
congr 3
funext j
rw [complexAverage_sum]
rfl
lemma complexGridAverage_differentiable : Differentiable ℂ (complexGridAverage bits ρ) := by
unfold complexGridAverage
apply Differentiable.fun_sum
intro g _
apply Differentiable.fun_smul
· unfold complexGridWeight
apply Differentiable.fun_finsetProd
intro j _
apply Differentiable.fun_finsetProd
intro y _
unfold complexLineLaw
fun_prop
· fun_prop
lemma complexGridAverage_norm_le_one [Nontrivial V] (z : ℂ)
(hρ : ∀g v, ‖ρ g v‖=‖v‖)
(hline : ∀j : Fin b,
∀σ : Representation ℂ (Equiv.Perm (Slot (bits j))) V,
(∀g v, ‖σ g v‖=‖v‖) → ‖complexAverage σ (complexLineLaw (bits j) z)‖ ≤ 1) :
‖complexGridAverage bits ρ z‖ ≤ 1 := by
rw [complexGridAverage_product]
apply reverse_product_norm_le_one
intro j
exact pi_comp_average_norm_le_one (I := GridOutside bits j)
(G := Equiv.Perm (Slot (bits j))) (H := Equiv.Perm (GridSlot bits)) (V := V)
ρ (gridStageHom bits j)
(complexLineLaw (bits j) z) hρ (hline j)
theorem complex_grid_disk (m : ℕ) : ∃ R : ℝ, 1 < R ∧
∀ (b : ℕ) (bits : Fin b → ℕ), (∀ j, bits j ≤ m) →
∀ (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] [Nontrivial V],
∀ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) V,
(∀g v, ‖ρ g v‖=‖v‖) → ∀z : ℂ, ‖z‖ ≤ R →
‖complexGridAverage bits ρ z‖ ≤ 1 := by
obtain ⟨R,hR,h⟩ := binary_lines_disk m
refine ⟨R,hR,?_⟩
intro b bits hm V _ _ _ _ ρ hρ z hz
apply complexGridAverage_norm_le_one bits ρ z hρ
intro j σ hσ
exact h (bits j) (hm j) V σ hσ z hz
end BinaryCoordinateSweeps
end
end OAI
|