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2f887db | 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 | import OAI.Computability.PerfectCompleteness.Decoding.LowerDirectionFiber
import OAI.Computability.PerfectCompleteness.Foundations.WholeArrayInteriorExteriorLemmas
namespace OAI
section
namespace PerfectCompleteness.LowerCutNodeCoordinates
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open WholeArraySubtreeSplit WholeArrayInteriorExterior
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
def scalarEquiv : {n m : Nat} → (p : Path branch n (m + 1)) →
(slots : Slots branch n → Fin t → MixedSupport.Slot) →
H (subtreeSlots p slots) ≃ₗ[F2] H (nodeSlots slots (upperNode p))
| _, _, .refl _, _ => LinearEquiv.refl F2 _
| _, _, .step i p, slots => scalarEquiv p (childSlots slots i)
def rowIndexEquiv (rows : Nat → Nat) : {n m : Nat} →
(p : Path branch n (m + 1)) →
Fin (rows (m + 1)) ≃ Fin (rows (Nodes.height (upperNode p)))
| _, _, .refl _ => Equiv.refl _
| _, _, .step _ p => rowIndexEquiv rows p
def directionEquiv (rows : Nat → Nat) : {n m : Nat} →
(p : Path branch n (m + 1)) →
BucketSampler.Direction (rows (m + 1)) ≃
BucketSampler.Direction (rows (Nodes.height (upperNode p)))
| _, _, .refl _ => Equiv.refl _
| _, _, .step _ p => directionEquiv rows p
def rowEquiv (rows : Nat → Nat) : {n m : Nat} →
(p : Path branch n (m + 1)) →
(slots : Slots branch n → Fin t → MixedSupport.Slot) →
(Fin (rows (m + 1)) → H (subtreeSlots p slots)) ≃ₗ[F2]
(Fin (rows (Nodes.height (upperNode p))) → H (nodeSlots slots (upperNode p)))
| _, _, .refl _, _ => LinearEquiv.refl F2 _
| _, _, .step i p, slots => rowEquiv rows p (childSlots slots i)
theorem directionEquiv_apply (rows : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(a : BucketSampler.Direction (rows (m + 1))) → (i : Fin (rows (m + 1))) →
(directionEquiv rows p a).val (rowIndexEquiv rows p i) = a.val i
| _, _, .refl _, _, _ => rfl
| _, _, .step _ p, a, i => directionEquiv_apply rows p a i
theorem rowEquiv_apply (rows : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(slots : Slots branch n → Fin t → MixedSupport.Slot) →
(S : Fin (rows (m + 1)) → H (subtreeSlots p slots)) →
(i : Fin (rows (m + 1))) →
rowEquiv rows p slots S (rowIndexEquiv rows p i) = scalarEquiv p slots (S i)
| _, _, .refl _, _, _, _ => rfl
| _, _, .step i p, slots, S, j => rowEquiv_apply rows p (childSlots slots i) S j
theorem pivot_transport (rows : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(a : BucketSampler.Direction (rows (m + 1))) →
rowIndexEquiv rows p (LowerDirectionFiber.pivot a) =
LowerDirectionFiber.pivot (directionEquiv rows p a)
| _, _, .refl _, _ => rfl
| _, _, .step _ p, a => pivot_transport rows p a
theorem rowEquiv_selectedRows (rows : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(slots : Slots branch n → Fin t → MixedSupport.Slot) →
(arrays : Arrays slots rows) →
rowEquiv rows p slots (SelectedArrayReplacement.selectedRows rows p slots arrays) =
arrays (upperNode p)
| _, _, .refl _, _, _ => rfl
| _, _, .step i p, slots, arrays =>
rowEquiv_selectedRows rows p (childSlots slots i) (fun node => arrays (.inr (i, node)))
theorem replace_eq_update (rows : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) (arrays : Arrays slots rows)
(fresh : Fin (rows (m + 1)) → H (subtreeSlots p slots)) :
SelectedArrayReplacement.replace rows p slots arrays fresh =
Function.update arrays (upperNode p) (rowEquiv rows p slots fresh) := by
funext node
by_cases hnode : node = upperNode p
· subst node
rw [Function.update_self]
have h := rowEquiv_selectedRows rows p slots
(SelectedArrayReplacement.replace rows p slots arrays fresh)
rw [SelectedArrayReplacement.selectedRows_replace] at h
exact h.symm
· rw [Function.update_of_ne hnode]
exact SelectedArrayReplacement.replace_outside rows p slots arrays fresh node hnode
theorem rowEquiv_replace (rows : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(slots : Slots branch n → Fin t → MixedSupport.Slot) →
(a : BucketSampler.Direction (rows (m + 1))) →
(S : Fin (rows (m + 1)) → H (subtreeSlots p slots)) →
(fresh : H (subtreeSlots p slots)) →
rowEquiv rows p slots (LowerDirectionFiber.replace a S fresh) =
LowerDirectionFiber.replace (directionEquiv rows p a)
(rowEquiv rows p slots S) (scalarEquiv p slots fresh)
| _, _, .refl _, _, _, _, _ => rfl
| _, _, .step i p, slots, a, S, fresh =>
rowEquiv_replace rows p (childSlots slots i) a S fresh
theorem replace_fiber_eq_update (rows : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) (arrays : Arrays slots rows)
(a : BucketSampler.Direction (rows (m + 1))) (fresh : H (subtreeSlots p slots)) :
SelectedArrayReplacement.replace rows p slots arrays
(LowerDirectionFiber.replace a
(SelectedArrayReplacement.selectedRows rows p slots arrays) fresh) =
Function.update arrays (upperNode p)
(LowerDirectionFiber.replace (directionEquiv rows p a)
(arrays (upperNode p)) (scalarEquiv p slots fresh)) := by
rw [replace_eq_update, rowEquiv_replace, rowEquiv_selectedRows]
end
end PerfectCompleteness.LowerCutNodeCoordinates
end
end OAI
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