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import OAI.Probability.BinarySweep.FiniteLaws.LinearEvenMoment
import OAI.Probability.BinarySweep.Conditional.ConditionalAverage
import OAI.Probability.BinarySweep.GridBounds.GridSplitSweep
namespace OAI
noncomputable section
section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.GridSplit
variable {m n h : ℕ} (bits : Fin (m+n) → ℕ) (H : PathFamily bits h)
@[simp] lemma left_join (x : GridSlot (leftBits bits)) (y : GridSlot (rightBits bits)) :
leftSlot bits (joinSlot bits x y)=x := by funext i; simp [leftSlot,joinSlot]
@[simp] lemma right_join (x : GridSlot (leftBits bits)) (y : GridSlot (rightBits bits)) :
rightSlot bits (joinSlot bits x y)=y := by funext i; simp [rightSlot,joinSlot]
@[simp] lemma join_left_right (x : GridSlot bits) :
joinSlot bits (leftSlot bits x) (rightSlot bits x)=x := (slotEquiv bits).symm_apply_apply x
def rowFreeEquiv (t : Fin (m+1)) : FreeSlot H (leftTime t) ≃
Σy : GridSlot (rightBits bits), FreeSlot (rowFamily bits H y) t where
toFun x := ⟨rightSlot bits x.val,⟨leftSlot bits x.val,by
rintro ⟨a,ha⟩
let k := (Fintype.equivFin (RowLabels bits H (rightSlot bits x.val))).symm a
apply x.property
refine ⟨k.val,?_⟩
apply (slotEquiv bits).injective
apply Prod.ext
· exact ha
· exact (rightSlot_leftTime bits H t k.val).trans k.property⟩⟩
invFun yx := ⟨joinSlot bits yx.2.val yx.1,by
rintro ⟨k,hk⟩
have he := congrArg (slotEquiv bits) hk
have hr : rightSlot bits (H.position 0 k)=yx.1 := by
rw [←rightSlot_leftTime bits H t k]
exact (congrArg Prod.snd he).trans (congrArg Prod.snd ((slotEquiv bits).apply_symm_apply (yx.2.val,yx.1)))
apply yx.2.property
refine ⟨Fintype.equivFin (RowLabels bits H yx.1) ⟨k,hr⟩,?_⟩
simp only [rowFamily,Equiv.symm_apply_apply]
exact (congrArg Prod.fst he).trans (congrArg Prod.fst ((slotEquiv bits).apply_symm_apply (yx.2.val,yx.1)))⟩
left_inv x := by
apply Subtype.ext
exact (slotEquiv bits).symm_apply_apply x.val
right_inv yx := by
rcases yx with ⟨y,x⟩
have he := (slotEquiv bits).apply_symm_apply (x.val,y)
apply Sigma.ext (congrArg Prod.snd he)
have hr : rightSlot bits (joinSlot bits x.val y)=y := congrArg Prod.snd he
apply (Subtype.heq_iff_coe_eq (by
intro a
change a∉Set.range ((rowFamily bits H (rightSlot bits (joinSlot bits x.val y))).position t) ↔ _
rw [hr])).mpr
exact congrArg Prod.fst he
def columnFreeEquiv (t : Fin (n+1)) : FreeSlot H (rightTime t) ≃
Σx : GridSlot (leftBits bits), FreeSlot (columnFamily bits H x) t where
toFun y := ⟨leftSlot bits y.val,⟨rightSlot bits y.val,by
rintro ⟨a,ha⟩
let k := (Fintype.equivFin (ColumnLabels bits H (leftSlot bits y.val))).symm a
apply y.property
refine ⟨k.val,?_⟩
apply (slotEquiv bits).injective
apply Prod.ext
· exact (leftSlot_rightTime bits H t k.val).trans k.property
· exact ha⟩⟩
invFun xy := ⟨joinSlot bits xy.1 xy.2.val,by
rintro ⟨k,hk⟩
have he := congrArg (slotEquiv bits) hk
have hl : leftSlot bits (H.position (rightTime (m:=m) 0) k)=xy.1 := by
rw [←leftSlot_rightTime bits H t k]
exact (congrArg Prod.fst he).trans (congrArg Prod.fst ((slotEquiv bits).apply_symm_apply (xy.1,xy.2.val)))
apply xy.2.property
refine ⟨Fintype.equivFin (ColumnLabels bits H xy.1) ⟨k,hl⟩,?_⟩
simp only [columnFamily,Equiv.symm_apply_apply]
exact (congrArg Prod.snd he).trans (congrArg Prod.snd ((slotEquiv bits).apply_symm_apply (xy.1,xy.2.val)))⟩
left_inv y := by
apply Subtype.ext
exact (slotEquiv bits).symm_apply_apply y.val
right_inv xy := by
rcases xy with ⟨x,y⟩
have he := (slotEquiv bits).apply_symm_apply (x,y.val)
apply Sigma.ext (congrArg Prod.fst he)
have hl : leftSlot bits (joinSlot bits x y.val)=x := congrArg Prod.fst he
apply (Subtype.heq_iff_coe_eq (by
intro a
change a∉Set.range ((columnFamily bits H (leftSlot bits (joinSlot bits x y.val))).position t) ↔ _
rw [hl])).mpr
exact congrArg Prod.snd he
abbrev RowSamples := ∀y, ConditionalChoices (rowFamily bits H y)
abbrev ColumnSamples := ∀x, ConditionalChoices (columnFamily bits H x)
def rowRemaining (g : RowSamples bits H) : FreeSlot H 0 ≃ FreeSlot H (rightTime (m:=m) 0) :=
(rowFreeEquiv bits H 0).trans ((Equiv.sigmaCongrRight (fun y =>
remainingBijection (rowFamily bits H y) (g y).val (g y).property)).trans
(rowFreeEquiv bits H (Fin.last m)).symm)
def columnRemaining (g : ColumnSamples bits H) :
FreeSlot H (rightTime (m:=m) 0) ≃ FreeSlot H (Fin.last (m+n)) :=
(columnFreeEquiv bits H 0).trans ((Equiv.sigmaCongrRight (fun x =>
remainingBijection (columnFamily bits H x) (g x).val (g x).property)).trans
(columnFreeEquiv bits H (Fin.last n)).symm)
@[simp] lemma rowRemaining_val (g : RowSamples bits H) (a : FreeSlot H 0) :
(rowRemaining bits H g a).val = joinSlot bits
(gridSweep (leftBits bits) (g (rightSlot bits a.val)).val (leftSlot bits a.val))
(rightSlot bits a.val) := rfl
@[simp] lemma columnRemaining_val (g : ColumnSamples bits H)
(a : FreeSlot H (rightTime (m:=m) 0)) :
(columnRemaining bits H g a).val = joinSlot bits (leftSlot bits a.val)
(gridSweep (rightBits bits) (g (leftSlot bits a.val)).val (rightSlot bits a.val)) := rfl
lemma remainingBijection_split (g : ConditionalChoices H) :
remainingBijection H g.val g.property =
(rowRemaining bits H ((conditionalChoicesEquiv bits H g).1)).trans
(columnRemaining bits H ((conditionalChoicesEquiv bits H g).2)) := by
apply Equiv.ext
intro a
apply Subtype.ext
change gridSweep bits g.val a.val =
(columnRemaining bits H ((conditionalChoicesEquiv bits H g).2)
(rowRemaining bits H ((conditionalChoicesEquiv bits H g).1) a)).val
rw [columnRemaining_val,rowRemaining_val,left_join,right_join]
apply (slotEquiv bits).injective
have he := Equiv.congr_fun (gridSweep_split bits g.val) (slotEquiv bits a.val)
simpa [conditionalChoicesEquiv,Equiv.permCongr_apply,rowPerm,columnPerm,
slotEquiv] using he
end BinaryCoordinateSweeps.GridSplit
end
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.GridSplit
open Irrep Representation
variable {m n h : ℕ} (bits : Fin (m+n) → ℕ) (H : PathFamily bits h)
def rowReference : FreeSlot H 0 ≃ FreeSlot H (rightTime (m:=m) 0) :=
(rowFreeEquiv bits H 0).trans ((Equiv.sigmaCongrRight (fun y =>
freeIdentification (rowFamily bits H y))).trans
(rowFreeEquiv bits H (Fin.last m)).symm)
def columnReference : FreeSlot H (rightTime (m:=m) 0) ≃ FreeSlot H (Fin.last (m+n)) :=
(columnFreeEquiv bits H 0).trans ((Equiv.sigmaCongrRight (fun x =>
freeIdentification (columnFamily bits H x))).trans
(columnFreeEquiv bits H (Fin.last n)).symm)
def rowRelative (g : RowSamples bits H) : Equiv.Perm (FreeSlot H 0) :=
(rowRemaining bits H g).trans (rowReference bits H).symm
def columnRelative (g : ColumnSamples bits H) : Equiv.Perm (FreeSlot H (rightTime (m:=m) 0)) :=
(columnRemaining bits H g).trans (columnReference bits H).symm
def framePermutation : Equiv.Perm (FreeSlot H 0) :=
((rowReference bits H).trans (columnReference bits H)).trans (freeIdentification H).symm
theorem remainingPerm_factor (g : ConditionalChoices H) :
remainingPerm H g.val g.property = framePermutation bits H *
(rowReference bits H).symm.permCongr
(columnRelative bits H ((conditionalChoicesEquiv bits H g).2)) *
rowRelative bits H ((conditionalChoicesEquiv bits H g).1) := by
unfold remainingPerm
rw [remainingBijection_split]
apply Equiv.ext
intro x
simp [framePermutation,rowRelative,columnRelative,Equiv.permCongr_apply]
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) V)
def rowAverage (z : ℝ) : V →ₗ[ℂ] V :=
∑g : RowSamples bits H,
((∏y, conditionalChoiceWeight (rowFamily bits H y) z (g y)):ℂ) • ρ (rowRelative bits H g)
def columnAverage (z : ℝ) : V →ₗ[ℂ] V :=
∑g : ColumnSamples bits H,
((∏x, conditionalChoiceWeight (columnFamily bits H x) z (g x)):ℂ) •
ρ ((rowReference bits H).symm.permCongr (columnRelative bits H g))
omit [FiniteDimensional ℂ V] in
theorem groupAverage_split (z : ℝ) :
groupAverage ρ (fun a => (conditionalGroupLaw H z a:ℂ)) =
ρ (framePermutation bits H) * columnAverage bits H ρ z * rowAverage bits H ρ z := by
rw [groupAverage_conditional,← Equiv.sum_comp (conditionalChoicesEquiv bits H).symm]
simp_rw [conditionalChoiceWeight_split,remainingPerm_factor,
Equiv.apply_symm_apply,map_mul,Complex.ofReal_mul]
rw [Fintype.sum_prod_type]
unfold columnAverage rowAverage
simp only [Finset.mul_sum,Finset.sum_mul,smul_mul_assoc,mul_smul_comm,
Finset.smul_sum,smul_smul,Complex.ofReal_prod]
theorem conditional_evenMoment_split (z : ℝ) (q : ℕ)
(hρ : ∀g v, ‖ρ g v‖=‖v‖) :
evenMoment q (groupAverage ρ (fun a => (conditionalGroupLaw H z a:ℂ))) =
evenMoment q (columnAverage bits H ρ z * rowAverage bits H ρ z) := by
rw [groupAverage_split,mul_assoc]
apply evenMoment_unitary_left
rw [unitary_rep_adjoint ρ hρ,←map_mul,inv_mul_cancel,map_one]
end BinaryCoordinateSweeps.GridSplit
end
end OAI
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