openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /EdmondsKarpAugmentation.lean
| import OAI.Combinatorics.ContingencyTables.Transport.EdmondsKarpDistance | |
| import OAI.Combinatorics.ContingencyTables.Transport.PathAugmentation | |
| namespace OAI | |
| /-! # Label monotonicity and critical arcs of a shortest augmentation -/ | |
| namespace ContingencyTables.EdmondsKarpAugmentation | |
| open PublishedEdmondsKarp PathAugmentation EdmondsKarpDistance | |
| open scoped Classical | |
| noncomputable section | |
| variable {n k : ℕ} | |
| lemma reverse_count_zero (p : Fin (k+1)→Fin n) (hp : Function.Injective p) (i : Fin k) : | |
| arcCount p (p i.succ) (p i.castSucc)=0 := by | |
| apply arcCount_eq_zero | |
| rintro ⟨j,hj,hi⟩ | |
| have h1 := congrArg Fin.val (hp hj) | |
| have h2 := congrArg Fin.val (hp hi) | |
| simp only [Fin.val_castSucc,Fin.val_succ] at h1 h2 | |
| omega | |
| lemma update_decrease (f : Flow n) (p : Fin (k+1)→Fin n) (a : ℤ) (ha : 0 ≤ a) | |
| (u v : Fin n) (h : update f p a u v < f u v) : | |
| ∃ i : Fin k,p i.castSucc=v ∧ p i.succ=u := by | |
| by_contra hn | |
| have hz : arcCount p v u=0 := arcCount_eq_zero p v u hn | |
| have hc := arcCount_nonneg p u v | |
| have hprod := mul_nonneg ha hc | |
| simp only [update,hz,sub_zero] at h | |
| omega | |
| lemma update_new_edge (c : Capacity n) (f : Flow n) (p : Fin (k+1)→Fin n) | |
| (a : ℤ) (ha : 0 ≤ a) (u v : Fin n) | |
| (h : 0 < (c u v:ℤ)-update f p a u v) : | |
| 0 < (c u v:ℤ)-f u v ∨ ∃ i : Fin k,p i.castSucc=v ∧ p i.succ=u := by | |
| by_cases hold : 0 < (c u v:ℤ)-f u v | |
| · exact Or.inl hold | |
| · exact Or.inr (update_decrease f p a ha u v (by omega)) | |
| lemma update_saturates (c : Capacity n) (f : Flow n) (p : Fin (k+1)→Fin n) | |
| (hp : Function.Injective p) (a : ℤ) (i : Fin k) | |
| (ha : a=(c (p i.castSucc) (p i.succ):ℤ)-f (p i.castSucc) (p i.succ)) : | |
| update f p a (p i.castSucc) (p i.succ)=c (p i.castSucc) (p i.succ) := by | |
| rw [update,arcCount_eq_one p hp i,reverse_count_zero p hp i] | |
| omega | |
| /-- All old distance labels remain valid lower potentials for the new | |
| residual graph, including arcs created by cancellation of flow. -/ | |
| theorem labels_monotone {c : Capacity n} {s t : Fin n} {f g : Flow n} | |
| (h : ShortestAugmentation c s t f g) (v : Fin n) : label c f s v ≤ label c g s v := by | |
| obtain ⟨k,p,a,hp,hshort,ha,_,_,hg⟩ := h | |
| have he : g=update f p a := funext (fun u => funext (hg u)) | |
| apply potential_le_distance (ResidualSearch.edge c g) s (label c f s) | |
| (label_source c f s) (label_le c f s) | |
| intro u v huv | |
| have hres : 0 < (c u v:ℤ)-g u v := of_decide_eq_true huv | |
| rw [he] at hres | |
| rcases update_new_edge c f p a ha.le u v hres with hold|⟨i,hi,hj⟩ | |
| · exact label_edge hold | |
| · have h1 := label_path hp hshort i.castSucc | |
| have h2 := label_path hp hshort i.succ | |
| rw [hi] at h1 | |
| rw [hj] at h2 | |
| simp only [Fin.val_castSucc,Fin.val_succ] at h1 h2 | |
| omega | |
| /-- A decrease of flow on (u,v) means the augmentation used (v,u). | |
| Consequently u was one distance layer beyond v. -/ | |
| theorem decrease_labels {c : Capacity n} {s t : Fin n} {f g : Flow n} | |
| (h : ShortestAugmentation c s t f g) (u v : Fin n) (hdec : g u v < f u v) : | |
| label c f s u=label c f s v+1 := by | |
| obtain ⟨k,p,a,hp,hshort,ha,_,_,hg⟩ := h | |
| have he : g=update f p a := funext (fun u => funext (hg u)) | |
| rw [he] at hdec | |
| obtain ⟨i,hi,hj⟩ := update_decrease f p a ha.le u v hdec | |
| have h1 := label_path hp hshort i.castSucc | |
| have h2 := label_path hp hshort i.succ | |
| rw [hi] at h1 | |
| rw [hj] at h2 | |
| simp only [Fin.val_castSucc,Fin.val_succ] at h1 h2 | |
| omega | |
| /-- Each augmentation has an arc which was residual, becomes saturated, | |
| and points from one finite distance layer to the next. -/ | |
| theorem critical_arc {c : Capacity n} {s t : Fin n} {f g : Flow n} | |
| (h : ShortestAugmentation c s t f g) : | |
| ∃ u v : Fin n,f u v < (c u v:ℤ) ∧ g u v=(c u v:ℤ) ∧ | |
| label c f s v=label c f s u+1 ∧ label c f s u < n := by | |
| obtain ⟨k,p,a,hp,hshort,ha,_,⟨i,hsat⟩,hg⟩ := h | |
| have he : g=update f p a := funext (fun u => funext (hg u)) | |
| refine ⟨p i.castSucc,p i.succ,?_,?_,?_,?_⟩ | |
| · have hr := hp.2.2.2 i | |
| omega | |
| · rw [he] | |
| exact update_saturates c f p hp.2.2.1 a i hsat | |
| · rw [label_path hp hshort i.castSucc,label_path hp hshort i.succ] | |
| rfl | |
| · exact label_path_lt hp hshort i.castSucc | |
| end | |
| end ContingencyTables.EdmondsKarpAugmentation | |
| end OAI | |