openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Selection /CandidateSpecificationTheorems.lean
| import OAI.Probability.MatroidSecretary.Selection.CandidateSpecification | |
| import OAI.Probability.MatroidSecretary.Selection.CandidateContracts | |
| namespace OAI | |
| /-! Correspondence of the independent source definitions with the implemented | |
| candidate filter and positive priority-greedy, followed by the source contract. -/ | |
| namespace MatroidProphet.CandidateSpecification | |
| open Finset | |
| lemma higher_eq_main {n : ℕ} (f : Fin n) (a : Option ℤ) | |
| (e : Fin n) (b : Option ℤ) : | |
| higher f a e b ↔ MainAlgorithm.higher f a e b := Iff.rfl | |
| lemma candidates_eq_main {n : ℕ} (M : Matroid (Fin n)) | |
| (a : Fin n → Option ℤ) (H : Finset (Fin n)) : | |
| candidates M a H = MainAlgorithm.candidateLabels M a H := by | |
| classical | |
| rw [MainAlgorithm.candidateLabels, MainAlgorithm.candidateSet_eq_filter] | |
| ext element | |
| by_cases hpositive : a element = none | |
| · simp [candidates, hpositive] | |
| · have hprior : {label | label ∈ H ∧ higher label (a label) element (a element)} = | |
| Candidates.prior (Priority.time a) H element := by | |
| ext label | |
| simp only [Candidates.prior, Set.mem_ofPred_eq, higher_eq_main, | |
| Priority.main_higher_iff_time a label element hpositive] | |
| simp [candidates, Candidates.candidates, hpositive, hprior] | |
| lemma greedy_eq_main {n : ℕ} (M : Matroid (Fin n)) | |
| (a : Fin n → Option ℤ) : greedy M a = MainAlgorithm.positiveGreedy M a := by | |
| classical | |
| ext e | |
| by_cases ha : a e = none | |
| · simp [greedy, MainAlgorithm.positiveGreedy, ha] | |
| · have hp : {f | higher f (a f) e (a e)} = | |
| Candidates.prior (Priority.time a) Finset.univ e := by | |
| ext f | |
| simp only [Candidates.prior, Finset.mem_univ, true_and, | |
| Set.mem_ofPred_eq, higher_eq_main, Priority.main_higher_iff_time a f e ha] | |
| simp [greedy, MainAlgorithm.positiveGreedy, Candidates.greedy, ha, hp] | |
| lemma survivors_eq_main {n : ℕ} (M : Matroid (Fin n)) | |
| (a : Fin n → Option ℤ) (H : Finset (Fin n)) : | |
| survivors M a H = MainAlgorithm.positiveGreedy M a \ H := by | |
| rw [survivors, greedy_eq_main] | |
| theorem candidateMass : CandidateMass := by | |
| intro n M hE w | |
| simpa only [survivors_eq_main, candidates_eq_main] using | |
| CandidateContracts.candidate_mass M hE w | |
| /-- Candidate membership uses only the focal key and keys on the revealed mask, | |
| never another unobserved label's key. -/ | |
| lemma candidate_membership_congr {n : ℕ} (M : Matroid (Fin n)) | |
| (a a' : Fin n → Option ℤ) (H : Finset (Fin n)) (e : Fin n) | |
| (he : a e = a' e) (hH : ∀ f ∈ H, a f = a' f) : | |
| e ∈ candidates M a H ↔ e ∈ candidates M a' H := by | |
| classical | |
| have hp : {f | f ∈ H ∧ higher f (a f) e (a e)} = | |
| {f | f ∈ H ∧ higher f (a' f) e (a' e)} := by | |
| ext f | |
| by_cases hf : f ∈ H | |
| · simp [hf, he, hH f hf] | |
| · simp [hf] | |
| simp only [candidates, Finset.mem_filter, Finset.mem_univ, true_and] | |
| rw [hp, he] | |
| /-- The source greedy definition really attains rounded OPT, with the source's | |
| rounding comparison on both sides. This verifies the ancillary input `I*` rather | |
| than assuming its optimality. -/ | |
| theorem source_rounding_and_greedy {n : ℕ} (M : Matroid (Fin n)) | |
| (hE : M.E = Set.univ) (w : Weights n) (hw : ∀ e, 0 ≤ w e) : | |
| let u := fun e => roundedWeight weightBase (w e) | |
| let I := greedy M (fun e => roundedLevel weightBase (w e)) | |
| optimum M w / weightBase ≤ optimum M u ∧ | |
| optimum M u ≤ optimum M w ∧ | |
| M.Indep (I : Set (Fin n)) ∧ | |
| (∑ e ∈ I, u e) = optimum M u := by | |
| simpa only [greedy_eq_main] using CandidateContracts.rounding_and_greedy M hE w hw | |
| end MatroidProphet.CandidateSpecification | |
| end OAI | |