| import Mathlib |
| import OAI.Probability.BinarySweep.Conditional.ConditionalSpecht |
| import OAI.Probability.BinarySweep.FiniteLaws.RemainingSign |
| import OAI.Probability.BinarySweep.SparseBounds.SparseSignGeometry |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
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| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
|
|
| section |
| attribute [local instance] Classical.propDecidable |
| variable {b h h' k : ℕ} {bits : Fin b → ℕ} {H : PathFamily bits h} {H' : PathFamily bits h'} |
|
|
| lemma PathExtension.signed_restrict_zero (E : PathExtension H H') (z : ℝ) |
| (hs : (∑g : GridChoices bits, if pathEvent H' g then |
| gridWeight bits z g*realSign (gridSweep bits g) else 0)=0) : |
| (∑g : ConditionalChoices H, if pathEvent H' g.val then |
| conditionalChoiceWeight H z g*realSign (remainingPerm H g.val g.property) else 0)=0 := by |
| obtain ⟨g₀,hg₀⟩ := pathEvent_nonempty H |
| let a := realSign (remainingPerm H g₀ hg₀)/realSign (gridSweep bits g₀) |
| have hs' : (∑g : ConditionalChoices H', gridWeight bits z g.val*realSign (gridSweep bits g.val))=0 := by |
| have he := sum_dite_zero (pathEvent H') |
| (fun g _ => gridWeight bits z g*realSign (gridSweep bits g)) |
| rw [←hs] |
| refine Eq.trans ?_ he.symm |
| apply Finset.sum_congr |
| · ext g; simp only [Finset.mem_univ] |
| intro g _; rfl |
| calc |
| _ = (a/conditionalNormalizer H z)* |
| (∑g : ConditionalChoices H, if pathEvent H' g.val then |
| gridWeight bits z g.val*realSign (gridSweep bits g.val) else 0) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro g _ |
| by_cases hg : pathEvent H' g.val |
| · rw [ite_eq_left hg,ite_eq_left hg,remaining_realSign_factor H g ⟨g₀,hg₀⟩] |
| dsimp [a,conditionalChoiceWeight] |
| ring |
| · simp only [ite_eq_right hg,mul_zero] |
| _ = (a/conditionalNormalizer H z)* |
| (∑g : ConditionalChoices H', gridWeight bits z g.val*realSign (gridSweep bits g.val)) := by |
| congr 1 |
| exact E.sum_restrict (fun g => gridWeight bits z g.val*realSign (gridSweep bits g.val)) |
| _ = 0 := by rw [hs',mul_zero] |
|
|
| variable (H) |
| def signedPlacementEntry (z : ℝ) (x : Placement H k 0) (y : Placement H k (Fin.last b)) : ℝ := |
| ∑g : ConditionalChoices H, if placementBijection H k g x=y then |
| conditionalChoiceWeight H z g*realSign (remainingPerm H g.val g.property) else 0 |
|
|
| lemma signedPlacementEntry_zero (z : ℝ) (x : Placement H k 0) (y : Placement H k (Fin.last b)) |
| (hs : ∀g₀ : ConditionalChoices H, |
| (∑g : GridChoices bits, if pathEvent (augmentByChoice H x g₀) g then |
| gridWeight bits z g*realSign (gridSweep bits g) else 0)=0) : |
| signedPlacementEntry H z x y=0 := by |
| by_cases he : ∃g₀ : ConditionalChoices H, placementBijection H k g₀ x=y |
| · obtain ⟨g₀,rfl⟩ := he |
| have hz := (augmentByChoice_extension H x g₀).signed_restrict_zero z (hs g₀) |
| refine Eq.trans ?_ hz |
| unfold signedPlacementEntry |
| apply Finset.sum_congr rfl |
| intro g _ |
| simp only [placementEvent_iff_augmentation] |
| · unfold signedPlacementEntry |
| apply Finset.sum_eq_zero |
| intro g _ |
| exact ite_eq_right (fun hg => he ⟨g,hg⟩) |
|
|
| theorem signedPlacementEntry_zero_many (z : ℝ) (j : Fin b) (hj : 0<bits j) |
| (hh : h+k<Fintype.card (GridOutside bits j)) |
| (x : Placement H k 0) (y : Placement H k (Fin.last b)) : |
| signedPlacementEntry H z x y=0 := |
| signedPlacementEntry_zero H z x y (fun g₀ => |
| signed_path_weight_zero_sparse (augmentByChoice H x g₀) z j hj hh) |
|
|
| theorem signedPlacementEntry_zero_few (j : Fin b) |
| (hh : h+k+Fintype.card (GridOutside bits j)<gridSize bits) |
| (x : Placement H k 0) (y : Placement H k (Fin.last b)) : |
| signedPlacementEntry H 0 x y=0 := |
| signedPlacementEntry_zero H 0 x y (fun g₀ => |
| signed_path_weight_zero_few (augmentByChoice H x g₀) j hh) |
|
|
| end |
|
|
| open Young Irrep |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
|
|
| lemma signed_partialKernel_conditional (z : ℝ) (A : Finset (Fin k)) |
| (x y : Placement H k 0) : |
| (∑a : Equiv.Perm (FreeSlot H 0), (conditionalGroupLaw H z a*realSign a:ℂ)*partialKernel A a y x)= |
| (signedPlacementEntry H z (restrictPlacement H x A) |
| (restrictPlacement H (placementIdentification H k y) A):ℂ) := by |
| have he := conditionalGroupLaw_expectation H z (fun a => (realSign a:ℂ)*partialKernel A a y x) |
| simp only [Complex.real_smul] at he |
| simp only [mul_assoc] |
| rw [he] |
| unfold signedPlacementEntry |
| rw [Complex.ofReal_sum] |
| apply Finset.sum_congr |
| · ext g; simp only [Finset.mem_univ] |
| intro g _ |
| simp only [partialKernel, partialKernel_event H g A x y, restrictPlacement_event] |
| split_ifs <;> simp |
| |
| lemma signed_centeredMatrix_zero (z : ℝ) |
| (hh : ∀ l ≤ k, ∀x : Placement H l 0, ∀y : Placement H l (Fin.last b), |
| signedPlacementEntry H z x y=0) : |
| centeredMatrix (I:=Fin k) (gridSize bits) |
| (fun a => (conditionalGroupLaw H z a*realSign a:ℂ))=0 := by |
| ext y x |
| simp only [centeredMatrix,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul,Matrix.zero_apply] |
| apply Finset.sum_eq_zero |
| intro A _ |
| rw [signed_partialKernel_conditional H z A x y] |
| have hA : Fintype.card A ≤ k := by simpa using Finset.card_le_univ A |
| rw [hh _ hA,Complex.ofReal_zero,mul_zero] |
|
|
| theorem conditional_signed_specht_one_le_zero (μ : YoungDiagram) |
| (e : Cell μ ≃ FreeSlot H 0) (a : Tail μ ≃ Fin k) (z : ℝ) |
| (hh : ∀ l ≤ k, ∀x : Placement H l 0, ∀y : Placement H l (Fin.last b), |
| signedPlacementEntry H z x y=0) : |
| evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun g => (conditionalGroupLaw H z g*realSign g:ℂ))) ≤ 0 := by |
| rw [groupAverage_comp_slots] |
| have hs : (gridSize bits:ℝ)≠0 := by exact_mod_cast (gridSize_pos bits).ne' |
| have he := specht_moment_one_le_centered_matrix μ (gridSize bits) hs |
| (fun g => (conditionalGroupLaw H z (e.permCongr g)*realSign (e.permCongr g):ℂ)) |
| have hr := centeredMatrix_hs_reindex a e (gridSize bits) |
| (fun g => (conditionalGroupLaw H z (e.permCongr g)*realSign (e.permCongr g):ℂ)) |
| simp only [Equiv.apply_symm_apply] at hr |
| rw [←hr,signed_centeredMatrix_zero H z hh] at he |
| simpa using he |
|
|
| theorem conditional_signed_specht_many (μ : YoungDiagram) |
| (e : Cell μ ≃ FreeSlot H 0) (a : Tail μ ≃ Fin k) (z : ℝ) |
| (j : Fin b) (hj : 0 < bits j) (hh : h+k < Fintype.card (GridOutside bits j)) : |
| evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun g => (conditionalGroupLaw H z g*realSign g:ℂ))) ≤ 0 := |
| conditional_signed_specht_one_le_zero H μ e a z (fun l hl x y => |
| signedPlacementEntry_zero_many H z j hj (by omega) x y) |
|
|
| theorem conditional_signed_specht_few (μ : YoungDiagram) |
| (e : Cell μ ≃ FreeSlot H 0) (a : Tail μ ≃ Fin k) |
| (j : Fin b) (hh : h+k+Fintype.card (GridOutside bits j) < gridSize bits) : |
| evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun g => (conditionalGroupLaw H 0 g*realSign g:ℂ))) ≤ 0 := |
| conditional_signed_specht_one_le_zero H μ e a 0 (fun l hl x y => |
| signedPlacementEntry_zero_few H j (by omega) x y) |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| open scoped BigOperators Classical |
| open Filter Topology |
|
|
| namespace BinaryCoordinateSweeps |
| open Young Irrep |
|
|
| variable {b h : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
|
|
| def signedCenteredHS (k : ℕ) (z : ℝ) : ℝ := ∑x : Placement H k 0, ∑y : Placement H k 0, |
| ‖centeredMatrix (gridSize bits) (fun a => (conditionalGroupLaw H z a*realSign a:ℂ)) y x‖^2 |
|
|
| lemma continuousAt_signedCenteredHS (k : ℕ) : ContinuousAt (signedCenteredHS H k) 0 := by |
| unfold signedCenteredHS |
| apply tendsto_finsetSum |
| intro x _ |
| apply tendsto_finsetSum |
| intro y _ |
| apply ContinuousAt.pow |
| apply ContinuousAt.norm |
| simp only [centeredMatrix,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul] |
| apply tendsto_finsetSum |
| intro A _ |
| apply ContinuousAt.mul continuousAt_const |
| apply tendsto_finsetSum |
| intro g _ |
| have hc := Complex.continuous_ofReal.continuousAt.comp (continuousAt_conditionalGroupLaw H g) |
| exact (hc.mul continuousAt_const).mul continuousAt_const |
|
|
| lemma signedCenteredHS_zero (k : ℕ) (z : ℝ) |
| (hh : ∀ l ≤ k, ∀x : Placement H l 0, ∀y : Placement H l (Fin.last b), |
| signedPlacementEntry H z x y=0) : signedCenteredHS H k z=0 := by |
| unfold signedCenteredHS |
| rw [signed_centeredMatrix_zero H z hh] |
| simp |
|
|
| lemma conditional_signed_specht_hs {k : ℕ} (μ : YoungDiagram) |
| (e : Cell μ ≃ FreeSlot H 0) (a : Tail μ ≃ Fin k) (z : ℝ) : |
| evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun g => (conditionalGroupLaw H z g*realSign g:ℂ))) ≤ signedCenteredHS H k z := by |
| rw [groupAverage_comp_slots] |
| have hs : (gridSize bits:ℝ)≠0 := by exact_mod_cast (gridSize_pos bits).ne' |
| have he := specht_moment_one_le_centered_matrix μ (gridSize bits) hs |
| (fun g => (conditionalGroupLaw H z (e.permCongr g)*realSign (e.permCongr g):ℂ)) |
| have hr := centeredMatrix_hs_reindex a e (gridSize bits) |
| (fun g => (conditionalGroupLaw H z (e.permCongr g)*realSign (e.permCongr g):ℂ)) |
| simp only [Equiv.apply_symm_apply] at hr |
| rwa [←hr] at he |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
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| end OAI |
|
|
