| import OAI.Probability.MatroidProphet.Maximum.Basic | |
| namespace OAI | |
| namespace MatroidProphet.Maximum | |
| open Finset | |
| variable {n : ℕ} | |
| lemma expectation_mem (q : Fin n → ℝ) (V : Finset (Fin n)) | |
| (e : Fin n) (he : e ∈ V) (c : ℝ) : | |
| bitsExpectation q V (fun H => if e ∈ H then c else 0) = q e * c := by | |
| have hno : bitsExpectation q (V.erase e) (fun H => if e ∈ H then c else 0) = 0 := by | |
| calc | |
| _ = bitsExpectation q (V.erase e) (fun _ => 0) := by | |
| apply bitsExpectation_congr | |
| intro H hH | |
| have heH : e ∉ H := fun hh => (Finset.mem_erase.mp (hH hh)).1 rfl | |
| simp [heH] | |
| _ = 0 := bitsExpectation_const q _ 0 | |
| have hyes : bitsExpectation q (V.erase e) (fun H => if e ∈ insert e H then c else 0) = c := by | |
| simp only [Finset.mem_insert_self, ite_true] | |
| exact bitsExpectation_const q _ c | |
| calc | |
| _ = bitsExpectation q (insert e (V.erase e)) (fun H => if e ∈ H then c else 0) := by | |
| rw [Finset.insert_erase he] | |
| _ = q e * c := by | |
| rw [bitsExpectation_insert q _ e (by simp), hno, hyes] | |
| ring | |
| lemma expectation_top_two_event (V : Finset (Fin n)) (g h : Fin n) | |
| (hg : g ∈ V) (hh : h ∈ V) (hgh : g ≠ h) (c : ℝ) : | |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V | |
| (fun H => if g ∈ H then 0 else if h ∈ H then c else 0) = c / 4 := by | |
| let q : Fin n → ℝ := fun _ => 1 / 2 | |
| have hno : bitsExpectation q (V.erase g) | |
| (fun H => if g ∈ H then 0 else if h ∈ H then c else 0) = (1 / 2 : ℝ) * c := by | |
| calc | |
| _ = bitsExpectation q (V.erase g) (fun H => if h ∈ H then c else 0) := by | |
| apply bitsExpectation_congr | |
| intro H hH | |
| have hgH : g ∉ H := fun hm => (Finset.mem_erase.mp (hH hm)).1 rfl | |
| simp [hgH] | |
| _ = _ := expectation_mem q (V.erase g) h (Finset.mem_erase.mpr ⟨hgh.symm, hh⟩) c | |
| have hyes : bitsExpectation q (V.erase g) | |
| (fun H => if g ∈ insert g H then 0 else if h ∈ insert g H then c else 0) = 0 := by | |
| simp only [Finset.mem_insert_self, ite_true] | |
| exact bitsExpectation_const q _ 0 | |
| change bitsExpectation q V _ = _ | |
| conv_lhs => rw [← Finset.insert_erase hg] | |
| rw [bitsExpectation_insert q _ g (by simp), hno, hyes] | |
| dsimp [q] | |
| ring | |
| theorem maximum_expectation_two (V : Finset (Fin n)) (g h : Fin n) | |
| (hg : g ∈ V) (hh : h ∈ V) (hgh : g ≠ h) | |
| (F : Finset (Fin n) → ℝ) (c : ℝ) | |
| (hF : ∀ H ⊆ V, 0 ≤ F H) | |
| (hsuccess : ∀ H ⊆ V, g ∉ H → h ∈ H → c ≤ F H) : | |
| c / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by | |
| rw [← expectation_top_two_event V g h hg hh hgh c] | |
| apply bitsExpectation_mono _ (fun _ => by norm_num) (fun _ => by norm_num) | |
| intro H hH | |
| split_ifs with hgH hhH | |
| · exact hF H hH | |
| · exact hsuccess H hH hgH hhH | |
| · exact hF H hH | |
| theorem maximum_expectation_one (V : Finset (Fin n)) (g : Fin n) (hg : g ∈ V) | |
| (F : Finset (Fin n) → ℝ) (c : ℝ) (hc : 0 ≤ c) | |
| (hF : ∀ H ⊆ V, 0 ≤ F H) | |
| (hsuccess : ∀ H ⊆ V, g ∉ H → c ≤ F H) : | |
| c / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by | |
| have hhalf : c / 2 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by | |
| have hevent := Candidates.expectation_not_mem (fun _ => (1 / 2 : ℝ)) V g hg c | |
| have hevent' : bitsExpectation (fun _ => (1 / 2 : ℝ)) V | |
| (fun H => if g ∉ H then c else 0) = c / 2 := by | |
| rw [hevent] | |
| ring | |
| rw [← hevent'] | |
| apply bitsExpectation_mono _ (fun _ => by norm_num) (fun _ => by norm_num) | |
| intro H hH | |
| by_cases hgH : g ∈ H | |
| · simpa only [hgH, not_true_eq_false, ite_false] using hF H hH | |
| · simpa only [hgH, not_false_eq_true, ite_true] using hsuccess H hH hgH | |
| linarith | |
| end MatroidProphet.Maximum | |
| end OAI | |