File size: 3,594 Bytes
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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Probability.SATComputability.SupercriticalProbability
import OAI.Probability.SATComputability.MaskDeletionTimes
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
open scoped Classical NNReal
theorem candidate_budget_survival {n m r : ℕ} (cs : Fin m → Fin 3 → SignedLiteral n) :
(candidateResidual (deletionBudgetMask n r) cs).Nonempty ↔
deletionSatisfiable r (fun j l => (cs j l).1) (fun j l => (cs j l).2) := by
rw [deletionSatisfiable_iff_candidate]
simp only [candidateResidual, deletionBudgetMask, Finset.filter_nonempty_iff,
Finset.mem_filter, Finset.mem_univ, true_and, not_forall, Prod.eta]
theorem candidate_zero_survival {n m : ℕ} (cs : Fin m → Fin 3 → SignedLiteral n) :
(candidateResidual (deletionBudgetMask n 0) cs).Nonempty ↔
(Finset.univ.filter (fun s : Assignment n => ∀ j, auxiliarySatisfies s (cs j))).Nonempty := by
rw [candidate_budget_survival]
simp only [Finset.filter_nonempty_iff, Finset.mem_univ, true_and]
constructor
· rintro ⟨D,hD,s,hs⟩
have hD0 : D = ∅ := Finset.card_eq_zero.mp (Nat.eq_zero_of_le_zero hD)
subst D
exact ⟨s,fun j => by simpa [auxiliarySatisfies, relaxedClauseSatisfied] using hs j⟩
· rintro ⟨s,hs⟩
exact ⟨∅,by simp,s,fun j => Or.inr (hs j)⟩
theorem finite_uniform_expect_eq_mean {A : Type*} [Fintype A] [Nonempty A] (f : A → ℝ) :
(FiniteLaw.uniform : FiniteLaw A).expect f = uniformMean f := by
simp only [FiniteLaw.expect, FiniteLaw.uniform, uniformMean,
div_eq_mul_inv, Finset.sum_mul, mul_comm]
theorem candidateStep_zero_survival {n : ℕ} [NeZero n] (m : ℕ) :
(candidateStep (n := n) 3)^[m] candidateAlive (deletionBudgetMask n 0) =
auxiliaryProbability n 3 m := by
rw [candidateStep_iterate, finite_uniform_expect_eq_mean]
unfold auxiliaryProbability auxiliarySurvival auxiliaryAverage embeddedAverage
rw [finiteClauseAverage_iterate]
apply uniformMean_congr
intro cs
unfold candidateAlive alive
congr 1
exact propext (candidate_zero_survival cs)
theorem candidateBlock_zero_survival {n : ℕ} [NeZero n] (r : ℝ≥0) :
(candidateBlock r 3 (deletionBudgetMask n 0)).expect candidateAlive =
∫ m : ℕ, auxiliaryProbability n 3 m ∂poissonMeasure r := by
rw [candidateBlock_expect]
simp only [candidateStep_zero_survival]
theorem candidate_full_nonempty {n m : ℕ} (cs : Fin m → Fin 3 → SignedLiteral n) :
(candidateResidual Finset.univ cs).Nonempty := by
refine ⟨fun _ => none, ?_⟩
simp only [candidateResidual, Finset.mem_filter, Finset.mem_univ, true_and]
intro j hj
have h := hj 0
simp [literalFalse] at h
theorem candidate_minimumDeletions_le {n m : ℕ} (cs : Fin m → RelaxedClause n) :
minimumDeletions (candidateResidual Finset.univ (fun j l => ((cs j).1 l, (cs j).2 l))) ≤
clauseMinimum cs := by
obtain ⟨s,hs⟩ := relaxedMinimum_attained (fun j => (cs j).1) (fun j => (cs j).2)
have hd := delete_one_per_violation s (fun j => (cs j).1) (fun j => (cs j).2)
change relaxedViolations s (fun j => (cs j).1) (fun j => (cs j).2) = clauseMinimum cs at hs
rw [hs] at hd
rw [deletionSatisfiable_iff_candidate] at hd
obtain ⟨x,hx,hxs⟩ := hd
apply (minimumDeletions_le_iff (candidate_full_nonempty _)).mpr
refine ⟨x, ?_, hx⟩
simp only [candidateResidual, Finset.mem_filter, Finset.mem_univ, true_and]
intro j hj
obtain ⟨l,hl⟩ := hxs j
exact hl (hj l)
end FixedClauseThreshold.Computability
end OAI
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