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a2ee19a | 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 | import OAI.Probability.SATComputability.CandidateSurvival
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass MeasureTheory ProbabilityTheory
open scoped NNReal Classical
theorem candidateResidual_inter {n k d : ℕ} (U : Finset (DeletionCandidate n))
(cs : Fin d → Fin k → SignedLiteral n) :
candidateResidual U cs = U ∩ candidateResidual Finset.univ cs := by
ext x
simp only [candidateResidual, Finset.mem_inter, Finset.mem_filter, Finset.mem_univ, true_and]
theorem candidateFixedBlock_mask {n : ℕ} [NeZero n] (k d : ℕ)
(U : Finset (DeletionCandidate n)) (f : Finset (DeletionCandidate n) → ℝ) :
(candidateFixedBlock k d Finset.univ).expect (fun V => f (U ∩ V)) =
(candidateFixedBlock k d U).expect f := by
simp only [candidateFixedBlock_expect, candidateStep_iterate]
apply FiniteLaw.expect_congr
intro cs
rw [candidateResidual_inter U cs]
theorem candidateBlock_mask {n : ℕ} [NeZero n] (r : ℝ≥0) (k : ℕ)
(U : Finset (DeletionCandidate n)) (f : Finset (DeletionCandidate n) → ℝ) :
(candidateBlock r k Finset.univ).expect (fun V => f (U ∩ V)) =
(candidateBlock r k U).expect f := by
simp only [candidateBlock, poissonMixture_expect, candidateFixedBlock_mask]
noncomputable def maskRun {n : ℕ} : (m : ℕ) → Finset (DeletionCandidate n) →
(Fin m → Finset (DeletionCandidate n)) → Finset (DeletionCandidate n)
| 0, U, _ => U
| m+1, U, xs => maskRun m (U ∩ xs 0) (Fin.tail xs)
theorem maskRun_law {n : ℕ} [NeZero n] (r : ℝ≥0) (k m : ℕ)
(U : Finset (DeletionCandidate n)) (f : Finset (DeletionCandidate n) → ℝ) :
(FiniteLaw.pi (fun _ : Fin m => candidateBlock r k Finset.univ)).expect
(fun xs => f (maskRun m U xs)) = (candidateBlock (r*m) k U).expect f := by
induction m generalizing U with
| zero =>
simp only [maskRun, FiniteLaw.expect_const, Nat.cast_zero, mul_zero]
rw [candidateBlock_expect, integral_poissonMeasure]
simp [zero_pow_eq, ite_div, ite_mul]
| succ m ih =>
rw [FiniteLaw.expect_pi_cons]
simp only [maskRun, Fin.cons_zero, Fin.tail_cons, ih]
rw [candidateBlock_mask r k U (fun V => (candidateBlock (r*m) k V).expect f)]
rw [← candidateBlock_comp]
congr 2
push_cast
ring
end FixedClauseThreshold.Computability
end OAI
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