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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import OAI.Computability.PerfectCompleteness.Construction.LocalCompletionFamily
import OAI.Computability.PerfectCompleteness.Foundations.CanonicalCoarseningErasure
namespace OAI
section
namespace PerfectCompleteness.CanonicalPartialTable
open MixedSupport CanonicalKeys CanonicalKeyShape CanonicalEdges CanonicalLabelEncoding
open scoped Classical
noncomputable section
abbrev Input (n k l : Nat) :=
Σ shapes : Fin n → Shape,
(Assignment (dummySlots shapes) → Fin k) × (Fin k → Fin l)
instance inputFintype (n k l : Nat) : Fintype (Input n k l) := by
unfold Input
infer_instance
variable {n k l q : Nat}
def slots (data : Input n k l) : Fin n → Slot := dummySlots data.1
def joint (data : Input n k l) : Assignment (slots data) → Fin k := data.2.1
def outputMap (data : Input n k l) : Fin k → Fin l := data.2.2
def leftKey (data : Input n k l) : Key n := key .left (slots data) (joint data)
def rightKey (data : Input n k l) : Key n :=
key .right (slots data) (outputMap data ∘ joint data)
theorem left_large (large : CanonicalKeyEncoding.partitionWidth n ≤ q) :
CanonicalKeyEncoding.partitionWidth n ≤ 2 * q := by omega
def leftEmbedding (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) : Label (slots data) (joint data) ↪ Fin (2 * q) :=
labelEmbedding (leftKey data) (left_large large)
def rightEmbedding (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) : Label (slots data) (outputMap data ∘ joint data) ↪ Fin q :=
labelEmbedding (rightKey data) large
def image (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) (a : Fin (2 * q)) : Fin q :=
if h : a ∈ Set.range (leftEmbedding large data) then
rightEmbedding large data
(coarsen (slots data) (joint data) (outputMap data) h.choose)
else ⟨0, hq⟩
def lookup (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) : LocalCompletionFamily.Input q :=
(paddedMask (leftKey data), image hq large data)
@[simp] theorem lookup_mask (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q) (data : Input n k l) :
(lookup hq large data).1 = paddedMask (leftKey data) := rfl
theorem mask_true (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) (a : Fin (2 * q)) :
(lookup hq large data).1 a = true ↔ a ∈ Set.range (leftEmbedding large data) :=
paddedMask_eq_true_iff (leftKey data) (left_large large) a
theorem image_legal (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) (P : Label (slots data) (joint data)) :
(lookup hq large data).2 (leftEmbedding large data P) =
rightEmbedding large data (coarsen (slots data) (joint data) (outputMap data) P) := by
have h : leftEmbedding large data P ∈ Set.range (leftEmbedding large data) := ⟨P, rfl⟩
have hchosen : h.choose = P := (leftEmbedding large data).injective h.choose_spec
change image hq large data (leftEmbedding large data P) = _
rw [image, dite_eq_left h, hchosen]
theorem image_illegal (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) (a : Fin (2 * q))
(hillegal : (lookup hq large data).1 a ≠ true) :
(lookup hq large data).2 a = ⟨0, hq⟩ := by
have h : a ∉ Set.range (leftEmbedding large data) := by
intro ha
exact hillegal ((mask_true hq large data a).2 ha)
exact dite_eq_right h
def legalEquiv (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) :
Label (slots data) (joint data) ≃ LocalCompletionFamily.Legal (lookup hq large data) where
toFun P := ⟨leftEmbedding large data P,
(mask_true hq large data _).2 ⟨P, rfl⟩⟩
invFun a := ((mask_true hq large data a.val).1 a.property).choose
left_inv P := (leftEmbedding large data).injective
((mask_true hq large data _).1
((mask_true hq large data _).2 ⟨P, rfl⟩)).choose_spec
right_inv a := Subtype.ext ((mask_true hq large data a.val).1 a.property).choose_spec
@[simp] theorem legalEquiv_val (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l) (P : Label (slots data) (joint data)) :
(legalEquiv hq large data P).val = leftEmbedding large data P := rfl
theorem admissible (hq : 0 < q) (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(data : Input n k l)
(hsmall : ∀ Q, Nat.card {P : Label (slots data) (joint data) //
coarsen (slots data) (joint data) (outputMap data) P = Q} ≤ 2) :
LocalCompletionFamily.Admissible (lookup hq large data) := by
let : Fintype (Label (slots data) (joint data)) := Fintype.ofFinite _
intro b
let fibers :
{P : Label (slots data) (joint data) // rightEmbedding large data
(coarsen (slots data) (joint data) (outputMap data) P) = b} ≃
{a : LocalCompletionFamily.Legal (lookup hq large data) //
(lookup hq large data).2 a.val = b} :=
(legalEquiv hq large data).subtypeEquiv (fun P => by
rw [legalEquiv_val, image_legal])
have hcard := Nat.card_congr fibers
have hbound := CompletionSoundness.LegalProjectionGame.embedded_fiber_le_two
(rightEmbedding large data) (coarsen (slots data) (joint data) (outputMap data))
(by intro Q; simpa only [← Nat.card_eq_fintype_card] using hsmall Q) b
simp only [← Nat.card_eq_fintype_card] at hcard hbound ⊢
exact hcard.symm.le.trans hbound
theorem admissible_of_output_fibers (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q) (data : Input n k l)
(hsmall : ∀ b, Nat.card {a : Fin k // outputMap data a = b} ≤ 2) :
LocalCompletionFamily.Admissible (lookup hq large data) :=
admissible hq large data
(coarsen_at_most_two (slots data) (joint data) (outputMap data) hsmall)
def input (sourceSlots : Fin n → Slot) (f : Assignment sourceSlots → Fin k)
(p : Fin k → Fin l) : Input n k l :=
⟨shape sourceSlots, transport sourceSlots f, p⟩
@[simp] theorem leftEmbedding_input (large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(sourceSlots : Fin n → Slot) (f : Assignment sourceSlots → Fin k)
(p : Fin k → Fin l) (P : Label sourceSlots f) :
leftEmbedding large (input sourceSlots f p)
(CanonicalCoarseningErasure.transportLabel sourceSlots f P) =
labelEmbedding (key .left sourceSlots f) (left_large large) P := rfl
theorem lookup_mask_actual (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(sourceSlots : Fin n → Slot) (f : Assignment sourceSlots → Fin k)
(p : Fin k → Fin l) :
(lookup hq large (input sourceSlots f p)).1 = paddedMask (key .left sourceSlots f) := by
funext a
change paddedMask (key .left (dummySlots (shape sourceSlots)) (transport sourceSlots f)) a = _
simp only [paddedMask, key, ← partition_erasure sourceSlots f]
theorem mask_actual_true (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(sourceSlots : Fin n → Slot) (f : Assignment sourceSlots → Fin k)
(p : Fin k → Fin l) (a : Fin (2 * q)) :
(lookup hq large (input sourceSlots f p)).1 a = true ↔
a ∈ Set.range (labelEmbedding (key .left sourceSlots f) (left_large large)) := by
rw [lookup_mask_actual]
exact paddedMask_eq_true_iff (key .left sourceSlots f) (left_large large) a
theorem image_actual_legal (hq : 0 < q)
(large : CanonicalKeyEncoding.partitionWidth n ≤ q)
(sourceSlots : Fin n → Slot) (f : Assignment sourceSlots → Fin k)
(p : Fin k → Fin l) (P : Label sourceSlots f) :
(lookup hq large (input sourceSlots f p)).2
(labelEmbedding (key .left sourceSlots f) (left_large large) P) =
labelEmbedding (key .right sourceSlots (p ∘ f)) large (coarsen sourceSlots f p P) := by
have h := image_legal hq large (input sourceSlots f p)
(CanonicalCoarseningErasure.transportLabel sourceSlots f P)
rw [leftEmbedding_input] at h
refine h.trans ?_
apply Fin.ext
change (CanonicalKeyEncoding.partEnum n
(coarsen (dummySlots (shape sourceSlots)) (transport sourceSlots f) p
(CanonicalCoarseningErasure.transportLabel sourceSlots f P)).val).val =
(CanonicalKeyEncoding.partEnum n (coarsen sourceSlots f p P).val).val
exact congrArg (fun a : Fin (CanonicalKeyEncoding.partitionWidth n) => a.val)
(CanonicalCoarseningErasure.partIndex_coarsen sourceSlots f p P).symm
end
end PerfectCompleteness.CanonicalPartialTable
end
end OAI
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