openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CellMinimumProgram.lean
| import OAI.Combinatorics.ContingencyTables.Machines.CellMaximumProgram | |
| import OAI.Combinatorics.ContingencyTables.Machines.BoundedComplementProgram | |
| namespace OAI | |
| /-! # Computing an attained minimum through the complementary table | |
| Complementation changes the lower endpoint query to the | |
| upper endpoint query and preserves every cell capacity. | |
| -/ | |
| namespace ContingencyTables.CellMinimumProgram | |
| open MatchingFPRAS.TreeTyped CellLowerTestProgram | |
| def complemented (z : Context) : Context := (BoundedComplementProgram.complement z.1,z.2) | |
| def complementedRealizer : Realizer complemented := | |
| pair (composition (f := fun z : Context=>z.1) | |
| (g := BoundedComplementProgram.complement) first BoundedComplementProgram.complementRealizer) second | |
| lemma polynomial_complemented : PolynomialTime complementedRealizer := | |
| polynomial_pair (polynomial_composition polynomial_first | |
| BoundedComplementProgram.polynomial_complement) polynomial_second | |
| def minimum (z : Context) : ℕ := | |
| CellMaximumProgram.bound z-CellMaximumProgram.maximum (complemented z) | |
| def minimumRealizer : Realizer minimum := | |
| composition (f := fun z : Context=> | |
| (CellMaximumProgram.bound z,CellMaximumProgram.maximum (complemented z))) | |
| (g := fun z : ℕ×ℕ=>z.1-z.2) | |
| (pair CellMaximumProgram.boundRealizer | |
| (composition (f := complemented) (g := CellMaximumProgram.maximum) | |
| complementedRealizer CellMaximumProgram.maximumRealizer)) subtract | |
| lemma polynomial_minimum : PolynomialTime minimumRealizer := | |
| polynomial_composition (polynomial_pair CellMaximumProgram.polynomial_bound | |
| (polynomial_composition polynomial_complemented CellMaximumProgram.polynomial_maximum)) | |
| polynomial_subtract | |
| lemma complemented_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) : | |
| complemented (NetworkVertexCode.data r c b,(i.val,j.val))= | |
| (NetworkVertexCode.data (BoundedTableComplement.rows b r) | |
| (BoundedTableComplement.columns b c) b,(i.val,j.val)) := by | |
| unfold complemented | |
| rw [BoundedComplementProgram.complement_code] | |
| /-- The returned lower endpoint is attained and is no larger than the | |
| selected coordinate of any feasible bounded table. -/ | |
| theorem minimum_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=minimum (NetworkVertexCode.data r c b,(i.val,j.val)) ∧ | |
| ∀ Y,HasMargins Y r c → (∀ u v,Y u v≤b u v) → X i j≤Y i j := by | |
| obtain ⟨X₀,hx,hxb⟩ := hne | |
| have hn : ∃ Y,HasMargins Y (BoundedTableComplement.rows b r) | |
| (BoundedTableComplement.columns b c) ∧ ∀ u v,Y u v≤b u v := | |
| ⟨BoundedTableComplement.complement b X₀, | |
| BoundedTableComplement.complement_margins b X₀ r c hx hxb, | |
| BoundedTableComplement.complement_bounds b X₀⟩ | |
| obtain ⟨Y,hy,hyb,hvalue,hmax⟩ := CellMaximumProgram.maximum_correct hEK | |
| (BoundedTableComplement.rows b r) (BoundedTableComplement.columns b c) b i j hn | |
| refine ⟨BoundedTableComplement.complement b Y, | |
| BoundedTableComplement.complement_reverse_margins b X₀ Y r c hx hxb hy hyb, | |
| BoundedTableComplement.complement_bounds b Y,?_,?_⟩ | |
| · unfold minimum | |
| rw [CellMaximumProgram.bound_code,complemented_code,←hvalue] | |
| rfl | |
| · intro Z hz hzb | |
| have hb := hmax (BoundedTableComplement.complement b Z) | |
| (BoundedTableComplement.complement_margins b Z r c hz hzb) | |
| (BoundedTableComplement.complement_bounds b Z) | |
| change b i j-Y i j≤Z i j | |
| change b i j-Z i j≤Y i j at hb | |
| omega | |
| end ContingencyTables.CellMinimumProgram | |
| end OAI | |