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135052a | 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.Probability.SATVariance.IsolatedSupports
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
lemma mem_take_ofFn.{u_1} {α : Type u_1} {M m : ℕ} (w : Fin M → α) (c : α) :
c ∈ (List.ofFn w).take m ↔ ∃ i : Fin M, i.val < m ∧ w i = c := by
rw [List.mem_take_iff_getElem]
constructor
· rintro ⟨j,hj,he⟩
have hj' : j < min m M := by simpa only [List.length_ofFn] using hj
refine ⟨⟨j,lt_of_lt_of_le hj' (min_le_right _ _)⟩,lt_of_lt_of_le hj' (min_le_left _ _),?_⟩
simpa only [List.getElem_ofFn] using he
· rintro ⟨i,hi,he⟩
refine ⟨i.val,?_,?_⟩
· simpa only [List.length_ofFn,lt_min_iff] using And.intro hi i.isLt
· simpa only [List.getElem_ofFn] using he
lemma listSAT_take_ofFn {n k M m : ℕ} (w : Fin M → Clause n k) :
listSAT ((List.ofFn w).take m) ↔ finiteSAT w m := by
constructor
· rintro ⟨a,ha⟩
exact ⟨a,fun i hi => ha (w i) ((mem_take_ofFn w _).mpr ⟨i,hi,rfl⟩)⟩
· rintro ⟨a,ha⟩
refine ⟨a,fun c hc => ?_⟩
obtain ⟨i,hi,rfl⟩ := (mem_take_ofFn w c).mp hc
exact ha i hi
lemma lt_listStop_ofFn_iff {n k M m : ℕ} (w : Fin M → Clause n k) (hm : m ≤ M) :
m < listStop (List.ofFn w) ↔ finiteSAT w m := by
rw [lt_listStop_iff,List.length_ofFn,and_iff_right hm,listSAT_take_ofFn]
lemma listStop_eq_finiteTime {n k M : ℕ} (w : Fin M → Clause n k) :
(listStop (List.ofFn w):ℝ) = finiteTime w + if finiteSAT w M then 1 else 0 := by
have hs : listStop (List.ofFn w) ≤ M+1 := by simpa only [List.length_ofFn] using listStop_le (List.ofFn w)
have hh (i : Fin (M+1)) : i.val < listStop (List.ofFn w) ↔ finiteSAT w i.val :=
lt_listStop_ofFn_iff w (by omega)
rw [← sum_fin_indicator_lt hs]
simp only [hh]
rw [Fin.sum_univ_castSucc]
rfl
lemma sentinel_error {n k M : ℕ} (w : Fin M → Clause n k) :
((listStop (List.ofFn w):ℝ)-finiteTime w)^2 ≤ 1 := by
rw [listStop_eq_finiteTime]
split_ifs <;> norm_num
lemma fvariance_bridge.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (f g : α → ℝ) :
fvariance f ≤ 2*fvariance g+2*favg (fun a => (f a-g a)^2) := by
apply (fvariance_le_sq_sub f (favg g)).trans
have hh := favg_mono (fun a : α => show (f a-favg g)^2 ≤ 2*(g a-favg g)^2+2*(f a-g a)^2 by
nlinarith [sq_nonneg ((g a-favg g)-(f a-g a))])
rw [favg_add,favg_mul,favg_mul,←fvariance_centered] at hh
exact hh
lemma sentinel_variance_bridge {n k M : ℕ} [Nonempty (Clause n k)] :
fvariance (fun w : Fin M → Clause n k => (listStop (List.ofFn w):ℝ)) ≤
2*fvariance (finiteTime (n := n) (k := k) (L := M))+2 := by
apply (fvariance_bridge _ _).trans
have hh := favg_mono (fun w : Fin M → Clause n k => sentinel_error w)
rw [favg_const] at hh
linarith
end RandomKSAT
end
end OAI
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