| import OAI.Probability.MatroidProphet.Algorithm.Feasible |
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| namespace OAI |
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| namespace MatroidProphet.MainAlgorithm |
| open Set Finset |
| variable {n : ℕ} |
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| lemma candidate_congr (M : Matroid (Fin n)) (d : MainMasks n) |
| (s s' : Fin n → Option ℤ) (h : ∀ e ∈ d.H, s e = s' e) (e : Fin n) (w : Option ℤ) : |
| candidate M d s e w ↔ candidate M d s' e w := by |
| have heq : {f | f ∈ d.H ∧ higher f (s f) e w} = {f | f ∈ d.H ∧ higher f (s' f) e w} := by |
| ext f |
| by_cases hf : f ∈ d.H |
| · simp only [Set.mem_ofPred_eq, hf, true_and, h f hf] |
| · simp [hf] |
| simp only [candidate, heq] |
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| lemma listed_congr (M : Matroid (Fin n)) (d : MainMasks n) |
| (s s' : Fin n → Option ℤ) |
| (hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e) : |
| listed M d s = listed M d s' := by |
| classical |
| unfold listed |
| apply Finset.biUnion_congr rfl |
| intro e he |
| have hc := candidate_congr M d s s' hH e (s e) |
| rw [hD e he] at hc |
| simp only [hD e he, hc] |
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| lemma groups_congr (M : Matroid (Fin n)) (d : MainMasks n) |
| (s s' : Fin n → Option ℤ) |
| (hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e) : |
| groups M d s = groups M d s' := by |
| rw [groups, groups, listed_congr M d s s' hH hD] |
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| lemma groupMask_congr (M : Matroid (Fin n)) (d : MainMasks n) |
| (s s' : Fin n → Option ℤ) (mask : Finset (Fin n)) |
| (hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e) |
| (hmask : ∀ e ∈ mask, s e = s' e) : |
| groupMask M d s mask = groupMask M d s' mask := by |
| funext j |
| ext e |
| by_cases he : e ∈ mask |
| · have hc := candidate_congr M d s s' hH e (s e) |
| rw [hmask e he] at hc |
| simp only [groupMask, Set.mem_ofPred_eq, he, true_and, |
| hmask e he, hc, groups_congr M d s s' hH hD] |
| · simp [groupMask, he] |
|
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| lemma preprocessing_congr (M : Matroid (Fin n)) (hE : M.E = Set.univ) (d : MainMasks n) |
| (s s' : Fin n → Option ℤ) (h : ∀ e ∈ d.H ∪ d.D ∪ d.C, s e = s' e) : |
| finalPath M hE d s = finalPath M hE d s' ∧ |
| birth M hE d s = birth M hE d s' ∧ assign M hE d s = assign M hE d s' := by |
| classical |
| have hH : ∀ e ∈ d.H, s e = s' e := fun e he => h e (mem_union_left _ (mem_union_left _ he)) |
| have hD : ∀ e ∈ d.D, s e = s' e := fun e he => h e (mem_union_left _ (mem_union_right _ he)) |
| have hC : ∀ e ∈ d.C, s e = s' e := fun e he => h e (mem_union_right _ he) |
| have hgd := groupMask_congr M d s s' d.D hH hD hD |
| have hgc := groupMask_congr M d s s' d.C hH hD hC |
| have hgs := groups_congr M d s s' hH hD |
| have hf : finalPath M hE d s = finalPath M hE d s' := by |
| simp only [finalPath, hgd, hgc, hgs] |
| have hb : birth M hE d s = birth M hE d s' := by |
| funext e i |
| simp only [birth, hgd, hgc, hgs] |
| refine ⟨hf, hb, ?_⟩ |
| funext e w |
| have hel (i : ℤ) : eligible M hE d s e i ↔ eligible M hE d s' e i := by |
| simp only [eligible, hf, hb, hgs, candidate_congr M d s s' hH] |
| cases w <;> simp only [assign, hel, hb] |
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| lemma observed_preprocessing (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (r : Seed (mainSeedBits n)) (w : Weights n) : |
| let s := fun e => roundedLevel weightBase (observed (hidden M hE) r w e) |
| let s' := fun e => roundedLevel weightBase (w e) |
| finalPath M hE (mainMasks r) s = finalPath M hE (mainMasks r) s' ∧ |
| birth M hE (mainMasks r) s = birth M hE (mainMasks r) s' ∧ |
| assign M hE (mainMasks r) s = assign M hE (mainMasks r) s' := by |
| apply preprocessing_congr |
| intro e he |
| unfold observed |
| have hm : e ∈ (hidden M hE).mask r := he |
| rw [ite_eq_left hm] |
|
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| end MatroidProphet.MainAlgorithm |
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| end OAI |
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