openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Sampling /BernoulliConditionalFiber.lean
| import OAI.Probability.MatroidProphet.Main | |
| namespace OAI | |
| /-! | |
| # Exact conditional fibers of a finite Bernoulli product law | |
| A fixed outside mask is a genuine conditioning event. Restricting the full | |
| product-law expectation to that event factors into its atom probability times | |
| the expectation over the unrevealed coordinates. The unnormalized identity | |
| also covers zero-probability fibers and endpoint Bernoulli rates. | |
| -/ | |
| namespace MatroidProphet | |
| open Finset | |
| variable {α : Type*} [DecidableEq α] | |
| /-- The weighted expectation of a single supported mask atom. -/ | |
| lemma bitsExpectation_mask_atom (q : α → ℝ) (V A : Finset α) | |
| (hA : A ⊆ V) (c : ℝ) : | |
| bitsExpectation q V (fun S => if S = A then c else 0) = | |
| bitsWeight q V A * c := by | |
| simp [bitsExpectation, mul_ite, hA] | |
| /-- Restrict a full product-law expectation to one outside-coordinate fiber. | |
| There are no positivity assumptions: null fibers are handled exactly. -/ | |
| theorem bitsExpectation_fixed_outside_event (q : α → ℝ) | |
| (V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G) | |
| (f : Finset α → ℝ) : | |
| bitsExpectation q V (fun T => if T \ G = A then f T else 0) = | |
| bitsWeight q (V \ G) A * | |
| bitsExpectation q G (fun B => f (A ∪ B)) := by | |
| rw [bitsExpectation_split q V G hG] | |
| have hf : bitsExpectation q (V \ G) | |
| (fun S => bitsExpectation q G | |
| (fun B => if (S ∪ B) \ G = A then f (S ∪ B) else 0)) = | |
| bitsExpectation q (V \ G) (fun S => | |
| if S = A then bitsExpectation q G (fun B => f (A ∪ B)) else 0) := by | |
| apply bitsExpectation_congr | |
| intro S hS | |
| have hout (B : Finset α) (hB : B ⊆ G) : (S ∪ B) \ G = S := by | |
| ext e | |
| simp only [mem_sdiff, mem_union] | |
| constructor | |
| · rintro ⟨heS | heB, heG⟩ | |
| · exact heS | |
| · exact False.elim (heG (hB heB)) | |
| · intro heS | |
| exact ⟨Or.inl heS, (mem_sdiff.mp (hS heS)).2⟩ | |
| by_cases hSA : S = A | |
| · subst S | |
| rw [ite_eq_left rfl] | |
| apply bitsExpectation_congr | |
| intro B hB | |
| rw [hout B hB, ite_eq_left rfl] | |
| · rw [ite_eq_right hSA] | |
| calc | |
| _ = bitsExpectation q G (fun _ => 0) := by | |
| apply bitsExpectation_congr | |
| intro B hB | |
| rw [hout B hB, ite_eq_right hSA] | |
| _ = 0 := bitsExpectation_const q G 0 | |
| rw [hf] | |
| exact bitsExpectation_mask_atom q (V \ G) A hA _ | |
| /-- The probability of a specified outside mask is exactly its product atom. -/ | |
| theorem bitsExpectation_fixed_outside_probability (q : α → ℝ) | |
| (V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G) : | |
| bitsExpectation q V (fun T => if T \ G = A then 1 else 0) = | |
| bitsWeight q (V \ G) A := by | |
| simpa only [bitsExpectation_const, mul_one] using | |
| bitsExpectation_fixed_outside_event q V G A hG hA (fun _ => 1) | |
| /-- On a nonnull outside fiber, the normalized conditional expectation is | |
| exactly the fresh-coordinate expectation used in the accounting contract. -/ | |
| theorem bitsExpectation_condition_on_outside (q : α → ℝ) | |
| (V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G) | |
| (hpos : bitsWeight q (V \ G) A ≠ 0) (f : Finset α → ℝ) : | |
| bitsExpectation q V (fun T => if T \ G = A then f T else 0) / | |
| bitsExpectation q V (fun T => if T \ G = A then 1 else 0) = | |
| bitsExpectation q G (fun B => f (A ∪ B)) := by | |
| rw [bitsExpectation_fixed_outside_event q V G A hG hA f, | |
| bitsExpectation_fixed_outside_probability q V G A hG hA] | |
| simp [hpos] | |
| /-- Interior coordinate rates make every outside atom positive, including | |
| empty/full masks and the empty outside ground set. -/ | |
| lemma bitsWeight_pos_of_interior (q : α → ℝ) (V A : Finset α) | |
| (hq0 : ∀ e ∈ V, 0 < q e) (hq1 : ∀ e ∈ V, q e < 1) : | |
| 0 < bitsWeight q V A := by | |
| unfold bitsWeight | |
| apply Finset.prod_pos | |
| intro e he | |
| split_ifs | |
| · exact hq0 e he | |
| · exact sub_pos.mpr (hq1 e he) | |
| /-- Constant interior-rate specialization: no nonnull-fiber premise remains. | |
| This applies in particular to the source's `t = 2⁻¹⁴⁰`. -/ | |
| theorem bitsExpectation_condition_on_outside_const | |
| (t : ℝ) (ht0 : 0 < t) (ht1 : t < 1) | |
| (V G A : Finset α) (hG : G ⊆ V) (hA : A ⊆ V \ G) | |
| (f : Finset α → ℝ) : | |
| bitsExpectation (fun _ => t) V (fun T => if T \ G = A then f T else 0) / | |
| bitsExpectation (fun _ => t) V (fun T => if T \ G = A then 1 else 0) = | |
| bitsExpectation (fun _ => t) G (fun B => f (A ∪ B)) := by | |
| exact bitsExpectation_condition_on_outside (fun _ => t) V G A hG hA | |
| (ne_of_gt (bitsWeight_pos_of_interior _ _ _ (fun _ _ => ht0) (fun _ _ => ht1))) f | |
| end MatroidProphet | |
| end OAI | |