File size: 5,851 Bytes
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.LowerCutScalarProjectionLemmas
namespace OAI
section
namespace PerfectCompleteness.LowerCutFormCollision
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
variable {branch : Nat → Nat} {n height t : Nat}
(path : Path branch n (height + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(inside : Slots branch (height + 1) → Fin t → MixedSupport.Slot)
(upper : Nodes branch n) (level : Nat)
(d : HierarchicalFrozenTables.LowerNodes upper level)
(hnode : WholeArrayInteriorExterior.upperNode path =
HierarchicalLeftDecoder.LowerNode upper level d)
def nativeScalar (f : H inside) :
HierarchicalDecoderTables.LowerH (CutSlotAssembly.fill path outside inside) upper level d :=
LowerCutDecoderForm.spaceEquiv path (CutSlotAssembly.fill path outside inside)
upper level d hnode
(CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill path outside inside).symm f)
def nativeForm
(answer : HierarchicalDecoderTables.Answer
(CutSlotAssembly.fill path outside inside) upper level d) :
ChildBilinearCollisionTransfer.ParentForm inside :=
CutNativeForms.formCast (CleanPhysicalReplay.cutSlots_fill path outside inside)
(LowerCutDecoderForm.form path (CutSlotAssembly.fill path outside inside)
upper level d hnode answer)
private theorem formCast_apply_inverse
{source target : Slots branch (height + 1) → Fin t → MixedSupport.Slot}
(h : source = target) (F : ChildBilinearCollisionTransfer.ParentForm source)
(f g : H target) :
CutNativeForms.formCast h F f g =
F (CutNativeForms.spaceCast h.symm f) (CutNativeForms.spaceCast h.symm g) := by
cases h
rfl
theorem nativeForm_apply
(answer : HierarchicalDecoderTables.Answer
(CutSlotAssembly.fill path outside inside) upper level d) (f g : H inside) :
nativeForm path outside inside upper level d hnode answer f g =
OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH
(CutSlotAssembly.fill path outside inside) upper level d) answer
(nativeScalar path outside inside upper level d hnode f)
(nativeScalar path outside inside upper level d hnode g) := by
unfold nativeForm
rw [formCast_apply_inverse]
rfl
variable (projected : Slots branch (height + 1) → Fin t → MixedSupport.Slot)
(projection : ∀ s k, MixedSupport.Projection (inside s k) (projected s k))
theorem nativeScalar_HPullback (f : H projected) :
nativeScalar path outside inside upper level d hnode (HPullback projection f) =
HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection path outside inside projected projection)
(HierarchicalLeftDecoder.LowerNode upper level d))
(nativeScalar path outside projected upper level d hnode f) :=
LowerCutScalarProjection.spaceEquiv_fill_HPullback path outside inside projected
upper level d hnode projection f
theorem restricted_on_native
(answer₀ answer₁ : Option (HierarchicalDecoderTables.Answer
(CutSlotAssembly.fill path outside inside) upper level d))
(U : Submodule F2 (H inside)) (hU : U ≤ LinearMap.range (HPullback projection))
(hcollision : BilinearCollisionTransfer.restrictedCollision
(answer₀.map (OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH
(CutSlotAssembly.fill path outside inside) upper level d)))
(answer₁.map (OddListExtraction.multiplicationForm
(HierarchicalDecoderTables.LowerH
(CutSlotAssembly.fill path outside inside) upper level d)))
(LinearMap.range (HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection path outside inside projected projection)
(HierarchicalLeftDecoder.LowerNode upper level d)))) = true) :
BilinearCollisionTransfer.restrictedCollision
(answer₀.map (nativeForm path outside inside upper level d hnode))
(answer₁.map (nativeForm path outside inside upper level d hnode)) U = true := by
obtain ⟨F, G, hF, hG, heq⟩ := of_decide_eq_true hcollision
cases answer₀ with
| none => cases hF
| some a₀ =>
cases answer₁ with
| none => cases hG
| some a₁ =>
have hF' := Option.some.inj hF
have hG' := Option.some.inj hG
rw [← hF', ← hG'] at heq
apply decide_eq_true
refine ⟨nativeForm path outside inside upper level d hnode a₀,
nativeForm path outside inside upper level d hnode a₁, rfl, rfl, ?_⟩
intro f hf g hg
rw [nativeForm_apply, nativeForm_apply]
obtain ⟨f', hf'⟩ := hU hf
obtain ⟨g', hg'⟩ := hU hg
have hfmem : nativeScalar path outside inside upper level d hnode f ∈
LinearMap.range (HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection path outside inside projected projection)
(HierarchicalLeftDecoder.LowerNode upper level d))) := by
refine ⟨nativeScalar path outside projected upper level d hnode f', ?_⟩
rw [← nativeScalar_HPullback, hf']
have hgmem : nativeScalar path outside inside upper level d hnode g ∈
LinearMap.range (HPullback (ChildBlockProjection.nodeProjection
(CutProjectionAssembly.fillProjection path outside inside projected projection)
(HierarchicalLeftDecoder.LowerNode upper level d))) := by
refine ⟨nativeScalar path outside projected upper level d hnode g', ?_⟩
rw [← nativeScalar_HPullback, hg']
exact heq _ hfmem _ hgmem
end
end PerfectCompleteness.LowerCutFormCollision
end
end OAI
|