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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import Mathlib
import OAI.Probability.BinarySweep.Analytic.ComplexGrid
import OAI.Probability.BinarySweep.Conditional.MomentOperatorBound
import OAI.Probability.BinarySweep.Conditional.MainMomentInterface
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
variable {b : ℕ} (bits : Fin b → ℕ)
def emptyPaths : PathFamily bits 0 where
position _ k := k.elim0
disjoint _ := fun k => k.elim0
changes_only_stage _ k := k.elim0
lemma emptyPaths_event (g : GridChoices bits) : pathEvent (emptyPaths bits) g := by
intro j k; exact k.elim0
def emptyFree (t : Fin (b+1)) : FreeSlot (emptyPaths bits) t ≃ GridSlot bits where
toFun := Subtype.val
invFun x := ⟨x,by rintro ⟨k,_⟩; exact k.elim0⟩
left_inv _ := rfl
right_inv _ := rfl
def emptyFrame : Equiv.Perm (GridSlot bits) :=
(emptyFree bits (Fin.last b)).symm.trans
((freeIdentification (emptyPaths bits)).symm.trans (emptyFree bits 0))
lemma emptyPerm (g : GridChoices bits) (hg : pathEvent (emptyPaths bits) g) :
(emptyFree bits 0).permCongr (remainingPerm (emptyPaths bits) g hg)=
emptyFrame bits * gridSweep bits g := by
ext x
rfl
lemma empty_normalizer (z : ℝ) : conditionalNormalizer (emptyPaths bits) z=1 := by
simp only [conditionalNormalizer,emptyPaths_event,ite_true,gridWeight_sum]
lemma empty_cost : pathCost (emptyPaths bits)=0 := by
have hh (j : Fin b) (y : GridOutside bits j) : lineHoles (emptyPaths bits) j y=0 := by
unfold lineHoles
exact Fintype.card_eq_zero
simp [pathCost,hh]
lemma empty_operator {D : ℕ}
(ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) (RepSpace D))
(z : ℝ) (x : RepSpace D) :
conditionalOperator (emptyPaths bits) z
(ρ.comp (emptyFree bits 0).permCongrHom.toMonoidHom) x=
ρ (emptyFrame bits) (complexGridAverage bits ρ (z:ℂ) x) := by
rw [conditionalOperator_apply,empty_normalizer]
simp only [Complex.ofReal_one,inv_one,one_smul,dite_true,emptyPaths_event]
simp only [complexGridAverage,sum_apply,smul_apply,
complexGridWeight_real,map_sum,map_smul]
apply Finset.sum_congr rfl
intro g _
congr 1
change ρ ((emptyFree bits 0).permCongr (remainingPerm _ g _)) x= _
rw [emptyPerm,map_mul]
rfl
lemma empty_operator_norm {D : ℕ}
(ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) (RepSpace D))
(hρ : IsUnitaryRep ρ) (z : ℝ) :
‖conditionalOperator (emptyPaths bits) z
(ρ.comp (emptyFree bits 0).permCongrHom.toMonoidHom)‖=
‖complexGridAverage bits ρ (z:ℂ)‖ := by
apply ContinuousLinearMap.opNorm_ext
intro x
rw [empty_operator,hρ]
theorem real_grid_bound : ∃ r : ℕ, 0 < r ∧ ∃ q : ℕ, 0 < q ∧
∃ a : ℝ, 0 < a ∧ ∀b : ℕ, 1 ≤ b → ∀bits : Fin b → ℕ,
(∀j, r ≤ bits j ∧ bits j ≤ 2*r) → ∀D : ℕ,
∀ρ : Representation ℂ (Equiv.Perm (GridSlot bits)) (RepSpace D),
ρ.IsIrreducible → IsUnitaryRep ρ → ∀z : ℝ, 0 ≤ z → z ≤ a →
‖complexGridAverage bits ρ (z:ℂ)‖ ≤ (D:ℝ)^(-c0/(2*q)) := by
obtain ⟨r,hr,q,hq,a,ha,h⟩ := conditional_moment_main
refine ⟨r,hr,q,hq,a,ha,?_⟩
intro b hb bits hd D ρ hi hρ z hz hza
let := hi
let σ : Representation ℂ (Equiv.Perm (FreeSlot (emptyPaths bits) 0)) (RepSpace D) := ρ.comp (emptyFree bits 0).permCongrHom.toMonoidHom
have hiσ : σ.IsIrreducible := Irrep.irreducible_comp_equiv _ ρ
have huσ : IsUnitaryRep σ := fun g x => hρ _ x
have hlog := h b hb bits hd 0 (emptyPaths bits) z hz hza D σ hiσ huσ
simp only [Nat.cast_zero,mul_zero,zero_mul,add_zero,empty_cost,sub_zero] at hlog
have hop := opNorm_of_logMoment hq _ hlog
rw [empty_operator_norm bits ρ hρ z] at hop
have hD : 0 < D := by
simpa only [RepSpace,finrank_euclideanSpace,Fintype.card_fin] using
Signed.irreducible_finrank_pos ρ
have hD' : 0 < (D:ℝ) := by exact_mod_cast hD
have hc : c0 ≤ cExponent (gridSize bits) := by
unfold cExponent
exact le_add_of_nonneg_right (by positivity)
have hlogD : 0 ≤ Real.log D := Real.log_nonneg (by exact_mod_cast hD)
have hqe : 0 < (2:ℝ)*q := by positivity
have hh : (-cExponent (gridSize bits)*Real.log D)/(2*q) ≤
(-c0/(2*q))*Real.log D := by
apply (div_le_iff₀ hqe).mpr
have := mul_le_mul_of_nonneg_right hc hlogD
field_simp
nlinarith
apply hop.trans
rw [Real.rpow_def_of_pos hD']
apply Real.exp_le_exp.mpr
simpa only [mul_comm] using hh
end BinaryCoordinateSweeps
end
end OAI
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