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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | import Mathlib
import OAI.Probability.BinarySweep.GridBounds.GridSplit
import OAI.Probability.BinarySweep.GridBounds.GridProjectedMoment
import OAI.Probability.BinarySweep.YoungTheory.HookHilbertOccurrence
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)
lemma left_path_address (j : Fin m) (k : Fin h) :
leftOutsideEquiv bits j (fun i => H.position (j.castAdd n).castSucc k i)=
(rightSlot bits (H.position 0 k),
fun i : {i : Fin m // i≠j} => leftSlot bits (H.position (leftTime j.castSucc) k) i) := by
apply Prod.ext
· change rightSlot bits (H.position (leftTime j.castSucc) k)=_
exact rightSlot_leftTime bits H j.castSucc k
· rfl
lemma right_path_address (j : Fin n) (k : Fin h) :
rightOutsideEquiv bits j (fun i => H.position (j.natAdd m).castSucc k i)=
(leftSlot bits (H.position (rightTime (m:=m) 0) k),
fun i : {i : Fin n // i≠j} => rightSlot bits (H.position (rightTime j.castSucc) k) i) := by
apply Prod.ext
· change leftSlot bits (H.position (rightTime j.castSucc) k)=_
exact leftSlot_rightTime bits H j.castSucc k
· rfl
lemma card_pair_fiber {K Y U : Type*} [Fintype K] {l : ℕ}
(f : K → Y) (g : K → U) (y : Y) (u : U)
(e : {k : K // f k=y} ≃ Fin l) :
Fintype.card {k : K // (f k,g k)=(y,u)} =
Fintype.card {i : Fin l // g (e.symm i).val=u} := by
apply Fintype.card_congr
refine { toFun := fun k => ⟨e ⟨k.val,congrArg Prod.fst k.property⟩, ?_⟩
invFun := fun i => ⟨(e.symm i.val).val, ?_⟩
left_inv := ?_
right_inv := ?_ }
· simpa only [Equiv.symm_apply_apply] using congrArg Prod.snd k.property
· exact Prod.ext (e.symm i.val).property i.property
· intro k; apply Subtype.ext; simp only [Equiv.symm_apply_apply]
· intro i; apply Subtype.ext
exact e.apply_symm_apply i.val
lemma lineHoles_left (j : Fin m) (u : GridOutside bits (j.castAdd n)) :
lineHoles H (j.castAdd n) u =
lineHoles (rowFamily bits H ((leftOutsideEquiv bits j) u).1) j
((leftOutsideEquiv bits j) u).2 := by
unfold lineHoles
let y := ((leftOutsideEquiv bits j) u).1
let v := ((leftOutsideEquiv bits j) u).2
have he (k : Fin h) : (fun i : {i : Fin (m+n) // i≠j.castAdd n} => H.position (j.castAdd n).castSucc k i)=u ↔
(rightSlot bits (H.position 0 k),
fun i : {i : Fin m // i≠j} => leftSlot bits (H.position (leftTime j.castSucc) k) i)=(y,v) := by
rw [← Equiv.apply_eq_iff_eq (leftOutsideEquiv bits j),left_path_address]
rw [Fintype.card_congr (Equiv.subtypeEquivRight he)]
convert card_pair_fiber (fun k => rightSlot bits (H.position 0 k))
(fun k => fun i : {i : Fin m // i≠j} => leftSlot bits (H.position (leftTime j.castSucc) k) i) y v
(@Fintype.equivFin (RowLabels bits H y) (rowLabelsFintype bits H y)) using 1 <;>
congr!
lemma lineHoles_right (j : Fin n) (u : GridOutside bits (j.natAdd m)) :
lineHoles H (j.natAdd m) u =
lineHoles (columnFamily bits H ((rightOutsideEquiv bits j) u).1) j
((rightOutsideEquiv bits j) u).2 := by
unfold lineHoles
let x := ((rightOutsideEquiv bits j) u).1
let v := ((rightOutsideEquiv bits j) u).2
have he (k : Fin h) : (fun i : {i : Fin (m+n) // i≠j.natAdd m} => H.position (j.natAdd m).castSucc k i)=u ↔
(leftSlot bits (H.position (rightTime (m:=m) 0) k),
fun i : {i : Fin n // i≠j} => rightSlot bits (H.position (rightTime j.castSucc) k) i)=(x,v) := by
rw [← Equiv.apply_eq_iff_eq (rightOutsideEquiv bits j),right_path_address]
rw [Fintype.card_congr (Equiv.subtypeEquivRight he)]
convert card_pair_fiber (fun k => leftSlot bits (H.position (rightTime (m:=m) 0) k))
(fun k => fun i : {i : Fin n // i≠j} => rightSlot bits (H.position (rightTime j.castSucc) k) i) x v
(@Fintype.equivFin (ColumnLabels bits H x) (columnLabelsFintype bits H x)) using 1 <;>
congr!
theorem pathCost_split : pathCost H =
(∑y, pathCost (rowFamily bits H y)) + (∑x, pathCost (columnFamily bits H x)) := by
unfold pathCost
rw [Fin.sum_univ_add]
congr 1
· rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro j _
simp_rw [lineHoles_left]
have he := Equiv.sum_comp (leftOutsideEquiv bits j) (fun yu =>
Real.log (((2^leftBits bits j:ℕ):ℝ)^lineHoles (rowFamily bits H yu.1) j yu.2 /
(Nat.descFactorial (2^leftBits bits j) (lineHoles (rowFamily bits H yu.1) j yu.2):ℝ)))
simpa only [Fintype.sum_prod_type] using he
· rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro j _
simp_rw [lineHoles_right]
have he := Equiv.sum_comp (rightOutsideEquiv bits j) (fun xu =>
Real.log (((2^rightBits bits j:ℕ):ℝ)^lineHoles (columnFamily bits H xu.1) j xu.2 /
(Nat.descFactorial (2^rightBits bits j) (lineHoles (columnFamily bits H xu.1) j xu.2):ℝ)))
simpa only [Fintype.sum_prod_type] using he
end BinaryCoordinateSweeps.GridSplit
end
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.GridSplit
open Irrep Representation Signed TraceHolder Density
variable {m n h : ℕ} (bits : Fin (m+n) → ℕ) (H : PathFamily bits h)
{A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
local instance : LinearOrder (GridSlot (rightBits bits)) := rowOrder bits
local instance : LinearOrder (GridSlot (leftBits bits)) := columnOrder bits
local instance : Nonempty (GridSlot (rightBits bits)) := ⟨fun _ _ => false⟩
def rowFreeSize (y : GridSlot (rightBits bits)) := Fintype.card (FreeSlot (rowFamily bits H y) 0)
def columnFreeSize (x : GridSlot (leftBits bits)) := Fintype.card (FreeSlot (columnFamily bits H x) 0)
lemma holeCount_junction :
(Fintype.card (GridSlot (rightBits bits)):ℝ)*Fintype.card (GridSlot (leftBits bits))-
(junctionSize bits H:ℝ)=h := by
have hh : h≤gridSize bits := by
have he := Fintype.card_le_of_injective (H.position 0) (H.disjoint 0)
simpa only [Fintype.card_fin,card_gridSlot] using he
rw [junctionSize_eq,Nat.cast_sub hh,card_gridSlot,card_gridSlot,
size_split,Nat.cast_mul]
ring
theorem conditional_dense_bound
{V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ (Equiv.Perm (Fin (junctionSize bits H))) V) [ρ.IsIrreducible]
(hρ : ∀g v, ‖ρ g v‖=‖v‖) (p : A → Bool)
(hocc : 0<Module.finrank ℂ (IntertwiningMap ρ (hilbertTensorRep p (junctionSize bits H))))
(z : ℝ) {q : ℕ} (hq : 0<q) (c : ℝ) (hc : 0≤c) (hcq : c≤q)
(ER : GridSlot (rightBits bits) → ℝ) (EC : GridSlot (leftBits bits) → ℝ)
(hchildR : ∀y (a : ActiveType p (rowFreeSize bits H y)),
evenMoment q (groupAverage (Young.partitionHilbertRep a.val) (rowChildWeight bits H z y))≤
Real.exp (-c*typeF a.val+ER y))
(hchildC : ∀x (a : ActiveType p (columnFreeSize bits H x)),
evenMoment q (groupAverage (Young.partitionHilbertRep a.val) (columnChildWeight bits H z x))≤
Real.exp (-c*typeF a.val+EC x)) :
evenMoment q (groupAverage (ρ.comp (inputToJunction bits H).permCongrHom.toMonoidHom)
(fun a => (conditionalGroupLaw H z a:ℂ)))≤
Real.exp (-c*Real.log (Module.finrank ℂ V:ℝ)+(∑y, ER y)+(∑x, EC x)+c*h+
(1+4*c+2*(q:ℝ))*(Fintype.card A:ℝ)^2*
(blockLogSize (rowFreeSize bits H)+blockLogSize (columnFreeSize bits H))) := by
refine (conditional_le_junction bits H ρ hρ p hocc z hq).trans ?_
have hb := grouped_dense_bound (physicalBoard bits H) p
(groupingOrder (rowGrouping bits H)) (groupingOrder (columnGrouping bits H))
(groupingPerm (rowGrouping bits H)) (groupingPerm (columnGrouping bits H))
(physicalBoard_row bits H) (physicalBoard_column bits H) ρ
(rowChildWeight bits H z) (columnChildWeight bits H z) hq c hc hcq ER EC hchildR hchildC
rw [holeCount_junction] at hb
exact hb
theorem conditional_hook_dense_bound {t : ℕ} (ht : 0<t)
(α : (junctionSize bits H).Partition) (hα : Young.InHook (Young.diagram α) t)
(z : ℝ) {q : ℕ} (hq : 0<q) (c : ℝ) (hc : 0≤c) (hcq : c≤q)
(ER : GridSlot (rightBits bits) → ℝ) (EC : GridSlot (leftBits bits) → ℝ)
(hchildR : ∀y (a : ActiveType (Prod.fst : Bool × Fin t → Bool) (rowFreeSize bits H y)),
evenMoment q (groupAverage (Young.partitionHilbertRep a.val) (rowChildWeight bits H z y))≤
Real.exp (-c*typeF a.val+ER y))
(hchildC : ∀x (a : ActiveType (Prod.fst : Bool × Fin t → Bool) (columnFreeSize bits H x)),
evenMoment q (groupAverage (Young.partitionHilbertRep a.val) (columnChildWeight bits H z x))≤
Real.exp (-c*typeF a.val+EC x)) :
evenMoment q (groupAverage ((Young.partitionHilbertRep α).comp
(inputToJunction bits H).permCongrHom.toMonoidHom)
(fun a => (conditionalGroupLaw H z a:ℂ)))≤
Real.exp (-c*typeF α+(∑y, ER y)+(∑x, EC x)+c*h+
(1+4*c+2*(q:ℝ))*(2*(t:ℝ))^2*
(blockLogSize (rowFreeSize bits H)+blockLogSize (columnFreeSize bits H))) := by
let : NeZero t := ⟨ne_of_gt ht⟩
have hb := conditional_dense_bound bits H (Young.partitionHilbertRep α)
(Young.partitionHilbertRep_unitary α) (Prod.fst : Bool × Fin t → Bool)
(Young.hookHilbert_occurs α hα) z hq c hc hcq ER EC hchildR hchildC
simpa only [Fintype.card_prod,Fintype.card_bool,Fintype.card_fin,Nat.cast_mul,Nat.cast_ofNat,typeF] using hb
end BinaryCoordinateSweeps.GridSplit
end
end OAI
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