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Twp strengthening adequacy #13

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Jan 24, 2024
Merged

Twp strengthening adequacy #13

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2347013
Add lemma on total_ub_lift and termination ineq
hei411 Jan 23, 2024
34f7954
strengthened adequacy!
hei411 Jan 24, 2024
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12 changes: 12 additions & 0 deletions theories/prob/union_bounds.v
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Expand Up @@ -912,4 +912,16 @@ Section total_ub_theory.
done.
Qed.

Lemma total_ub_lift_termination_ineq (μ : distr A) f ε:
total_ub_lift μ f ε -> 1 - ε <= SeriesC μ.
Proof.
intros H2.
assert (∀ x, Decision (f x)).
{ intros. apply make_decision. }
trans (prob μ (λ x, bool_decide (f x))).
- apply H2. intros. by apply bool_decide_eq_true_2.
- apply SeriesC_le; last apply pmf_ex_seriesC.
intros n; case_bool_decide; pose proof (pmf_pos (μ) n); lra.
Qed.

End total_ub_theory.
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