openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /BoundedComplementProgram.lean
| import OAI.Combinatorics.ContingencyTables.Transport.BoundedTableComplement | |
| import OAI.Combinatorics.ContingencyTables.Machines.NetworkFeasibilityProgram | |
| namespace OAI | |
| /-! # Computing complementary bounded-table margins in binary -/ | |
| namespace ContingencyTables.BoundedComplementProgram | |
| open MatchingFPRAS.TreeTyped Polynomial NetworkVertexCode | |
| abbrev EntryInput := Input×ℕ | |
| def rowValue (z : EntryInput) : ℕ := | |
| (z.1.2.getD z.2 []).sum-z.1.1.1.getD z.2 0 | |
| def rowValueRealizer : Realizer rowValue := | |
| composition (pair | |
| (composition (composition (pair (composition first second) second) (getDFast [])) listSum) | |
| (composition (pair (composition (composition first first) first) second) (getDFast 0))) subtract | |
| lemma polynomial_rowValue : PolynomialTime rowValueRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_second) polynomial_second) | |
| (polynomial_getDFast [])) polynomial_listSum) | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_first) | |
| polynomial_first) polynomial_second) (polynomial_getDFast 0))) polynomial_subtract | |
| def columnValues (z : EntryInput) : List ℕ := z.1.2.map (fun row=>row.getD z.2 0) | |
| def columnValuesRealizer : Realizer columnValues := | |
| composition (pair second (composition first second)) | |
| (mapWith (composition (pair second first) (getDFast 0))) | |
| lemma polynomial_columnValues : PolynomialTime columnValuesRealizer := | |
| polynomial_composition (polynomial_pair polynomial_second | |
| (polynomial_composition polynomial_first polynomial_second)) | |
| (polynomial_mapWith (polynomial_composition (polynomial_pair polynomial_second polynomial_first) | |
| (polynomial_getDFast 0))) | |
| def columnValue (z : EntryInput) : ℕ := | |
| (columnValues z).sum-z.1.1.2.getD z.2 0 | |
| def columnValueRealizer : Realizer columnValue := | |
| composition (pair (composition columnValuesRealizer listSum) | |
| (composition (pair (composition (composition first first) second) second) (getDFast 0))) subtract | |
| lemma polynomial_columnValue : PolynomialTime columnValueRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_columnValues polynomial_listSum) | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_first) | |
| polynomial_second) polynomial_second) (polynomial_getDFast 0))) polynomial_subtract | |
| def rowIndices (z : Input) : List ℕ := List.range z.1.1.length | |
| def columnIndices (z : Input) : List ℕ := List.range z.1.2.length | |
| def rowIndicesRealizer : Realizer rowIndices := | |
| composition (composition (composition first first) listLength) range | |
| def columnIndicesRealizer : Realizer columnIndices := | |
| composition (composition (composition first second) listLength) range | |
| lemma polynomial_rowIndices : PolynomialTime rowIndicesRealizer := by | |
| apply polynomial_boundedRange | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_first) | |
| polynomial_listLength) X | |
| intro z | |
| simpa only [Function.comp_apply,eval_X] using | |
| (list_length_le_weight z.1.1).trans ((weight_fst z.1).trans (weight_fst z)) | |
| lemma polynomial_columnIndices : PolynomialTime columnIndicesRealizer := by | |
| apply polynomial_boundedRange | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_listLength) X | |
| intro z | |
| simpa only [Function.comp_apply,eval_X] using | |
| (list_length_le_weight z.1.2).trans ((weight_snd z.1).trans (weight_fst z)) | |
| def complement (z : Input) : Input := | |
| (((rowIndices z).map (fun i=>rowValue (z,i)), | |
| (columnIndices z).map (fun j=>columnValue (z,j))),z.2) | |
| def complementRealizer : Realizer complement := | |
| pair (pair | |
| (composition (pair identity rowIndicesRealizer) (mapWith rowValueRealizer)) | |
| (composition (pair identity columnIndicesRealizer) (mapWith columnValueRealizer))) second | |
| lemma polynomial_complement : PolynomialTime complementRealizer := | |
| polynomial_pair (polynomial_pair | |
| (polynomial_composition (polynomial_pair polynomial_identity polynomial_rowIndices) | |
| (polynomial_mapWith polynomial_rowValue)) | |
| (polynomial_composition (polynomial_pair polynomial_identity polynomial_columnIndices) | |
| (polynomial_mapWith polynomial_columnValue))) polynomial_second | |
| lemma rowValue_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) (i : Fin m) : | |
| rowValue (data r c b,i.val)=BoundedTableComplement.rows b r i := by | |
| simp only [rowValue,data,ResidualGraphProgram.natCode,List.getD_eq_getElem?_getD, | |
| List.getElem?_ofFn,i.isLt,dite_true,Option.getD_some,Fin.eta,List.sum_ofFn, | |
| BoundedTableComplement.rows,BoundedTableComplement.rowTotal] | |
| lemma columnValue_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) (j : Fin n) : | |
| columnValue (data r c b,j.val)=BoundedTableComplement.columns b c j := by | |
| simp only [columnValue,columnValues,data,ResidualGraphProgram.natCode,List.map_ofFn, | |
| Function.comp_def,List.getD_eq_getElem?_getD,List.getElem?_ofFn,j.isLt, | |
| dite_true,Option.getD_some,Fin.eta,List.sum_ofFn, | |
| BoundedTableComplement.columns,BoundedTableComplement.columnTotal] | |
| lemma complement_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) : | |
| complement (data r c b)=data (BoundedTableComplement.rows b r) | |
| (BoundedTableComplement.columns b c) b := by | |
| unfold complement | |
| apply Prod.ext | |
| · apply Prod.ext | |
| · simp only [rowIndices,data,List.length_ofFn,DirectedBFSCode.range_eq_map,←List.ofFn_eq_map, | |
| List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext i | |
| exact rowValue_code r c b i | |
| · simp only [columnIndices,data,List.length_ofFn,DirectedBFSCode.range_eq_map,←List.ofFn_eq_map, | |
| List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext j | |
| exact columnValue_code r c b j | |
| · rfl | |
| end ContingencyTables.BoundedComplementProgram | |
| end OAI | |