File size: 3,135 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 | import OAI.Computability.PerfectCompleteness.Foundations.KeyFiniteness
import OAI.Computability.PerfectCompleteness.Machines.FiniteBlockMachineLemmas
namespace OAI
section
namespace PerfectCompleteness.CanonicalLabelEncoding
open CanonicalKeys CanonicalKeyEncoding
open scoped Classical
noncomputable section
def labelIndex {n : Nat} (key : Key n) : KeyLabel key ↪ Fin (partitionWidth n) where
toFun label := partEnum n label.val
inj' := by
intro a b h
exact Subtype.ext ((partEnum n).injective h)
def labelEmbedding {n q : Nat} (key : Key n) (large : partitionWidth n ≤ q) :
KeyLabel key ↪ Fin q :=
(labelIndex key).trans (Fin.castLEEmb large)
@[simp] theorem labelEmbedding_val {n q : Nat} (key : Key n)
(large : partitionWidth n ≤ q) (label : KeyLabel key) :
(labelEmbedding key large label).val = (partEnum n label.val).val := rfl
theorem labelIndex_eq_iff {n : Nat} {left right : Key n}
(a : KeyLabel left) (b : KeyLabel right) :
labelIndex left a = labelIndex right b ↔ a.val = b.val :=
(partEnum n).injective.eq_iff
def paddedMask {n q : Nat} (key : Key n) (a : Fin q) : Bool :=
if h : a.val < partitionWidth n then partitionMask key.partition ⟨a.val, h⟩ else false
theorem paddedMask_eq_true_iff {n q : Nat} (key : Key n)
(large : partitionWidth n ≤ q) (a : Fin q) :
paddedMask key a = true ↔ a ∈ Set.range (labelEmbedding key large) := by
by_cases h : a.val < partitionWidth n
· rw [paddedMask, dite_eq_left h]
change decide ((partEnum n).symm ⟨a.val, h⟩ ∈ key.partition) = true ↔ _
rw [decide_eq_true_iff]
constructor
· intro hpart
refine ⟨⟨(partEnum n).symm ⟨a.val, h⟩, hpart⟩, ?_⟩
apply Fin.ext
simp only [labelEmbedding_val, Equiv.apply_symm_apply]
· rintro ⟨label, heq⟩
have hi : partEnum n label.val = (⟨a.val, h⟩ : Fin (partitionWidth n)) := by
apply Fin.ext
exact congrArg (fun x : Fin q => x.val) heq
simpa only [← hi, Equiv.symm_apply_apply] using label.property
· rw [paddedMask, dite_eq_right h]
constructor
· simp
· rintro ⟨label, heq⟩
have hval : (partEnum n label.val).val = a.val := congrArg (fun x : Fin q => x.val) heq
exact (h (hval ▸ (partEnum n label.val).isLt)).elim
theorem label_card_le {n : Nat} (key : Key n) [Fintype (KeyLabel key)] :
Fintype.card (KeyLabel key) ≤ partitionWidth n := by
simpa only [Fintype.card_fin] using Fintype.card_le_of_injective (labelIndex key)
(labelIndex key).injective
theorem labelIndex_projection {n : Nat} {Y : Type*} (side : Side)
{slots projected : Fin n → MixedSupport.Slot}
(p : ∀ i, MixedSupport.Projection (slots i) (projected i))
(f : MixedSupport.Assignment projected → Y)
(label : CanonicalKeys.Label slots (f ∘ MixedSupport.projectionMap p)) :
labelIndex (CanonicalKeys.key side slots (f ∘ MixedSupport.projectionMap p)) label =
labelIndex (CanonicalKeys.key side projected f) (CanonicalKeys.transportLabel p f label) :=
rfl
end
end PerfectCompleteness.CanonicalLabelEncoding
end
end OAI
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