File size: 3,232 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 72 73 74 75 76 | import OAI.Combinatorics.EditApproximation.Probability.RejectionSelectedLaw
namespace OAI
/-! Literal rejection trials are charged only for source coordinates
encountered by the shared memo execution. The accepted coordinates may
influence the entire request list; failure counts remain independent. -/
namespace EditApproximation
open MeasureTheory ProbabilityTheory
section Memo
variable {σ κ : Type*} [Fintype σ] [DecidableEq σ] [Fintype κ] [DecidableEq κ]
(range : σ → ℕ) [∀ key, NeZero (range key)]
omit [∀ key, NeZero (range key)] in
theorem memo_selected_source_bits_le (keys : List (Sum σ κ)) (bits : ℕ)
(hbits : ∀ key, rejectionBitCount (range key) ≤ bits) :
(∑ key, (rejectionBitCount (range key) : ℝ) *
selectionIndicator (Sum.inl key ∈ keys)) ≤ keys.length * bits := by
let weight : Sum σ κ → ℕ := Sum.elim (fun key => rejectionBitCount (range key)) (fun _ => 0)
have hs := sum_selected_nat_weights_le keys weight
rw [Fintype.sum_sum_type] at hs
simp only [weight, Sum.elim_inl, Sum.elim_inr, Nat.cast_zero, mul_zero,
Finset.sum_const_zero, add_zero] at hs
have hb : (keys.map weight).sum ≤ keys.length * bits := by
clear hs
induction keys with
| nil => simp
| cons key keys ih =>
have hk : weight key ≤ bits := by cases key <;> simp only [weight, Sum.elim_inl, Sum.elim_inr]; exact hbits _; omega
simp only [List.map_cons, List.sum_cons, List.length_cons, Nat.add_mul]
omega
calc
_ = ∑ key, selectionIndicator (Sum.inl key ∈ keys) * (rejectionBitCount (range key) : ℝ) := by
apply Finset.sum_congr rfl
intro key _
ring
_ ≤ ((keys.map weight).sum : ℕ) := hs
_ ≤ (keys.length * bits : ℕ) := by exact_mod_cast hb
_ = _ := by push_cast; rfl
theorem memo_rejectionBits_expected_le
(freshKeys : (∀ key, Fin (range key)) → List (Sum σ κ))
(requests bits : ℕ)
(hrequests : ∀ accepted, (freshKeys accepted).length ≤ requests)
(hbits : ∀ key, rejectionBitCount (range key) ≤ bits) :
(∫ execution, rejectionSelectedBits range
(fun key accepted => Sum.inl key ∈ freshKeys accepted) execution
∂rejectionRuntimeArrayLaw range) ≤ 2 * (requests * bits : ℕ) := by
apply rejectionSelectedBits_expected_le
intro accepted
exact (memo_selected_source_bits_le range (freshKeys accepted) bits hbits).trans
(by exact_mod_cast Nat.mul_le_mul_right bits (hrequests accepted))
theorem memo_rejectionBits_expected_le_real
(freshKeys : (∀ key, Fin (range key)) → List (Sum σ κ))
(budget : ℝ) (bits : ℕ)
(hrequests : ∀ accepted, ((freshKeys accepted).length : ℝ) ≤ budget)
(hbits : ∀ key, rejectionBitCount (range key) ≤ bits) :
(∫ execution, rejectionSelectedBits range
(fun key accepted => Sum.inl key ∈ freshKeys accepted) execution
∂rejectionRuntimeArrayLaw range) ≤ 2 * (budget * bits) := by
apply rejectionSelectedBits_expected_le
intro accepted
exact (memo_selected_source_bits_le range (freshKeys accepted) bits hbits).trans
(mul_le_mul_of_nonneg_right (hrequests accepted) (Nat.cast_nonneg _))
end Memo
end EditApproximation
end OAI
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