File size: 3,774 Bytes
e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Binary.BinaryGroupSetupWork
import OAI.Combinatorics.EditApproximation.Binary.BinaryGradientWork
namespace OAI
/-! Group evaluation reads the exact finite sampler indexed by the count
computed from the stored rational scale. Casting indices below is proof
transport only; the emitted natural count is used by the executed loop. -/
namespace EditApproximation
namespace BinaryFraction
def countedGroupMemberWithWork {M T : ℕ} (Q b : ℕ) (scales : Fin T → BinaryFraction)
(connection : BinaryFraction) (envelope values : Vector BinaryFraction M)
(draw : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → Fin M) : BinaryFraction × ℕ :=
let counts := vectorMapWithWork T fun t => groupSampleCountWithWork M Q b (scales t)
let count := fun t => bitWordValue (counts.1.get t)
let heq : ∀ t, count t = groupSampleCount M Q b (scales t).value := by
intro t
simp only [count, counts, vectorMapWithWork_value, groupSampleCountWithWork_value]
let result := groupMemberWithWork scales count connection envelope values
(fun t j => draw t (Fin.cast (heq t) j))
(result.1, counts.2 + result.2 + 1)
theorem groupMemberWithWork_transport {M T : ℕ}
(scales : Fin T → BinaryFraction) (left right : Fin T → ℕ) (h : ∀ t, left t = right t)
(connection : BinaryFraction) (envelope values : Vector BinaryFraction M)
(draw : ∀ t, Fin (right t) → Fin M) :
groupMemberWithWork scales left connection envelope values (fun t j => draw t (Fin.cast (h t) j)) =
groupMemberWithWork scales right connection envelope values draw := by
have heq : left = right := funext h
subst right
rfl
@[simp] theorem countedGroupMemberWithWork_value {M T : ℕ} (Q b : ℕ) (scales : Fin T → BinaryFraction)
(connection : BinaryFraction) (envelope values : Vector BinaryFraction M)
(draw : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → Fin M) :
(countedGroupMemberWithWork Q b scales connection envelope values draw).1.value =
groupMemberEstimate (fun t => (scales t).value) (fun t => groupSampleCount M Q b (scales t).value)
connection.value (fun i => (envelope.get i).value) (fun i => (values.get i).value) draw := by
unfold countedGroupMemberWithWork
dsimp only
rw [groupMemberWithWork_transport]
exact groupMemberWithWork_value _ _ _ _ _ _
theorem countedGroupMemberWithWork_cost {M T : ℕ} (Q b : ℕ) (scales : Fin T → BinaryFraction)
(connection : BinaryFraction) (envelope values : Vector BinaryFraction M)
(draw : ∀ t, Fin (groupSampleCount M Q b (scales t).value) → Fin M)
(B C W : ℕ) (hM : Nat.size M + 2 ≤ B) (hQ : Nat.size Q + 2 ≤ B)
(hb : Nat.size b + 2 ≤ B) (hs : ∀ t, (scales t).width ≤ B)
(hconn : connection.width ≤ B)
(hc : ∀ t, (groupScaleWithWork (groupSampleCount M Q b (scales t).value) (scales t)
envelope values (draw t)).2 ≤ C)
(hw : ∀ t, (groupScaleWithWork (groupSampleCount M Q b (scales t).value) (scales t)
envelope values (draw t)).1.width ≤ W) :
(countedGroupMemberWithWork Q b scales connection envelope values draw).2 ≤
T * (groupSampleCountBitBudget B + 6) + groupMemberBitBudget T B C W + 1 := by
have hcounts := vectorMapWithWork_cost_le T (fun t => groupSampleCountWithWork M Q b (scales t))
(groupSampleCountBitBudget B) (fun t => (groupSampleCountWithWork_bounds M Q b (scales t) B hM hQ hb (hs t)).2)
have hmember := (groupMemberWithWork_bounds scales
(fun t => groupSampleCount M Q b (scales t).value) connection envelope values draw B C W hconn hc hw).2
unfold countedGroupMemberWithWork
dsimp only
rw [groupMemberWithWork_transport]
omega
end BinaryFraction
end EditApproximation
end OAI
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