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dd14c26 | 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 | import OAI.Probability.MatroidProphet.FiniteBits
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [DecidableEq α]
theorem bitsExpectation_restrict (q : α → ℝ) (W V : Finset α) (hV : V ⊆ W)
(f : Finset α → ℝ)
(hf : ∀ S S', (∀ e ∈ V, e ∈ S ↔ e ∈ S') → f S = f S') :
bitsExpectation q W f = bitsExpectation q V f := by
induction W using Finset.strongInductionOn with
| _ W ih =>
by_cases hW : W ⊆ V
· rw [subset_antisymm hW hV]
· obtain ⟨e, heW, heV⟩ := Finset.not_subset.mp hW
have hVerase : V ⊆ W.erase e := by
intro x hx
exact mem_erase.mpr ⟨fun h => heV (h ▸ hx), hV hx⟩
have heq : bitsExpectation q (W.erase e) (fun S => f (insert e S)) =
bitsExpectation q (W.erase e) f := by
apply bitsExpectation_congr
intro S _
apply hf
intro x hx
have hxe : x ≠ e := fun h => heV (h ▸ hx)
simp [hxe]
have hs := bitsExpectation_insert q (W.erase e) e (notMem_erase e W) f
rw [insert_erase heW, heq, ih (W.erase e) (erase_ssubset heW) hVerase] at hs
linarith
lemma bitsExpectation_sum {ι : Type*} (q : α → ℝ) (V : Finset α) (I : Finset ι)
(f : ι → Finset α → ℝ) :
bitsExpectation q V (fun S => ∑ e ∈ I, f e S) = ∑ e ∈ I, bitsExpectation q V (f e) := by
simp only [bitsExpectation, mul_sum]
rw [sum_comm]
lemma bitsExpectation_mem (q : α → ℝ) (V : Finset α) (e : α) (he : e ∈ V) :
bitsExpectation q V (fun S => if e ∈ S then 1 else 0) = q e := by
rw [bitsExpectation_restrict q V {e} (singleton_subset_iff.mpr he)
(fun S => if e ∈ S then 1 else 0) (by
intro S S' h
simp only [h e (mem_singleton_self e)])]
rw [show ({e} : Finset α) = insert e ∅ from rfl,
bitsExpectation_insert q ∅ e (notMem_empty e)]
simp [bitsExpectation_empty]
lemma bitsExpectation_card_inter (q : α → ℝ) (V K : Finset α) (hKV : K ⊆ V) :
bitsExpectation q V (fun S => ((K ∩ S).card : ℝ)) = ∑ e ∈ K, q e := by
have hsum (S : Finset α) : ((K ∩ S).card : ℝ) =
∑ e ∈ K, if e ∈ S then (1 : ℝ) else 0 := by
simp
simp_rw [hsum]
rw [bitsExpectation_sum]
exact sum_congr rfl (fun e he => bitsExpectation_mem q V e (hKV he))
end MatroidProphet
end OAI
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