openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CellMaximumProgram.lean
| import OAI.Combinatorics.ContingencyTables.Machines.CellLowerTestProgram | |
| import OAI.Combinatorics.ContingencyTables.Machines.BinarySearchProgram | |
| namespace OAI | |
| /-! # Computing an attained cell maximum by binary feasibility search | |
| A finite maximum is used only to prove the result of the executed binary | |
| search. The program never enumerates feasible tables. | |
| -/ | |
| namespace ContingencyTables.CellMaximumProgram | |
| open MatchingFPRAS.TreeTyped CellLowerTestProgram | |
| open scoped Classical | |
| def bound (z : Context) : ℕ := ResidualGraphProgram.capacityEntry (z.1.2,z.2) | |
| def boundRealizer : Realizer bound := | |
| composition (pair (composition first second) second) ResidualGraphProgram.capacityEntryRealizer | |
| lemma polynomial_bound : PolynomialTime boundRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_second) polynomial_second) | |
| ResidualGraphProgram.polynomial_capacityEntry | |
| def maximum (z : Context) : ℕ := BinarySearchProgram.search test (z,bound z) | |
| def maximumRealizer : Realizer maximum := | |
| composition (f := fun z : Context=>(z,bound z)) | |
| (g := BinarySearchProgram.search test) | |
| (pair identity boundRealizer) (BinarySearchProgram.searchRealizer test testRealizer) | |
| lemma polynomial_maximum : PolynomialTime maximumRealizer := | |
| polynomial_composition (polynomial_pair polynomial_identity polynomial_bound) | |
| (BinarySearchProgram.polynomial_search test testRealizer polynomial_test) | |
| lemma bound_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) : | |
| bound (NetworkVertexCode.data r c b,(i.val,j.val))=b i j := | |
| ResidualGraphProgram.capacityEntry_code b i j | |
| /-- For every feasible instance the computed value is attained, and it | |
| bounds the same coordinate of every other feasible table. -/ | |
| theorem maximum_correct {m n : ℕ} (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) | |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) | |
| (hne : ∃ X,HasMargins X r c ∧ ∀ u v,X u v≤b u v) : | |
| ∃ X,HasMargins X r c ∧ (∀ u v,X u v≤b u v) ∧ | |
| X i j=maximum (NetworkVertexCode.data r c b,(i.val,j.val)) ∧ | |
| ∀ Y,HasMargins Y r c → (∀ u v,Y u v≤b u v) → Y i j≤X i j := by | |
| let S : Finset (BoundedTable r c b) := Finset.univ | |
| have hS : S.Nonempty := by | |
| obtain ⟨X,hm,hb⟩ := hne | |
| exact ⟨⟨⟨X,hm⟩,hb⟩,Finset.mem_univ _⟩ | |
| obtain ⟨M,hM,hmax⟩ := Finset.exists_max_image S | |
| (fun X : BoundedTable r c b=>X.val.val i j) hS | |
| have ht (t : ℕ) : | |
| test ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔ t≤M.val.val i j := by | |
| rw [CellLowerTestProgram.test_correct hEK] | |
| constructor | |
| · rintro ⟨X,hm,hb,hx⟩ | |
| exact hx.trans (hmax ⟨⟨X,hm⟩,hb⟩ (Finset.mem_univ _)) | |
| · intro hx | |
| exact ⟨M.val.val,M.val.property,M.property,hx⟩ | |
| have he : maximum (NetworkVertexCode.data r c b,(i.val,j.val))=M.val.val i j := by | |
| unfold maximum | |
| rw [bound_code] | |
| exact BinarySearchProgram.search_eq test _ _ _ ht (M.property i j) | |
| exact ⟨M.val.val,M.val.property,M.property,he.symm, | |
| fun Y hm hb=>hmax ⟨⟨Y,hm⟩,hb⟩ (Finset.mem_univ _)⟩ | |
| end ContingencyTables.CellMaximumProgram | |
| end OAI | |