Expected Cost Nat eq sum tail probs of pathwise Cost At Most
AddWriterT.expectedCostNat_eq_sum_tail_probs_of_pathwiseCostAtMost
Plain-language statement
Finite tail-sum formula for natural-valued writer cost under a pathwise upper bound. If every execution path of oa incurs cost at most n, then the tail probabilities vanish above n, so the infinite tail sum truncates to Finset.range n.
Exact Lean statement
lemma expectedCostNat_eq_sum_tail_probs_of_pathwiseCostAtMost
[LawfulMonad m]
{oa : AddWriterT ℕ m α} {n : ℕ}
(h : PathwiseCostAtMost oa n) :
expectedCostNat oa = ∑ i ∈ Finset.range n, Pr[ fun c ↦ i < c | oa.costs ]Formal artifact
Lean source
lemma expectedCostNat_eq_sum_tail_probs_of_pathwiseCostAtMost [LawfulMonad m] {oa : AddWriterT ℕ m α} {n : ℕ} (h : PathwiseCostAtMost oa n) : expectedCostNat oa = ∑ i ∈ Finset.range n, Pr[ fun c ↦ i < c | oa.costs ] := by rw [expectedCostNat_eq_tsum_tail_probs] symm rw [tsum_eq_sum (s := Finset.range n) (fun b hb ↦ ?_)] · have hnb : n ≤ b := Nat.le_of_not_lt (by simpa [Finset.mem_range] using hb) refine probEvent_eq_zero fun c hc ↦ ?_ rw [AddWriterT.costs_def, support_map] at hc rcases hc with ⟨z, hz, rfl⟩ exact not_lt_of_ge (le_trans (h z hz) hnb)- Project
- VCVio
- License
- Apache-2.0
- Commit
- 2ceb2d825ee3
- Source
- VCVio/OracleComp/QueryTracking/WriterCost.lean:245-257
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