openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CellLowerTestProgram.lean
| import OAI.Combinatorics.ContingencyTables.Machines.LowerCapacityProgram | |
| import OAI.Combinatorics.MatchingCount.Machines.TreeFiniteOps | |
| namespace OAI | |
| /-! # The executed feasibility test with one extra lower cell bound -/ | |
| namespace ContingencyTables.CellLowerTestProgram | |
| open MatchingFPRAS.TreeTyped Polynomial | |
| abbrev Context := NetworkVertexCode.Input×(ℕ×ℕ) | |
| abbrev Input := Context×ℕ | |
| def shifted (z : Input) : NetworkVertexCode.Input := | |
| ((ListSubtractProgram.subtractAt ((z.1.1.1.1,z.1.2.1),z.2), | |
| ListSubtractProgram.subtractAt ((z.1.1.1.2,z.1.2.2),z.2)), | |
| LowerCapacityProgram.matrix ((z.1.1.2,z.1.2),z.2)) | |
| def rowInputRealizer : Realizer (fun z : Input=>((z.1.1.1.1,z.1.2.1),z.2)) := | |
| pair (pair (composition (composition (composition first first) first) first) | |
| (composition (composition first second) first)) second | |
| def columnInputRealizer : Realizer (fun z : Input=>((z.1.1.1.2,z.1.2.2),z.2)) := | |
| pair (pair (composition (composition (composition first first) first) second) | |
| (composition (composition first second) second)) second | |
| def capacityInputRealizer : Realizer (fun z : Input=>((z.1.1.2,z.1.2),z.2)) := | |
| pair (pair (composition (composition first first) second) (composition first second)) second | |
| lemma polynomial_rowInput : PolynomialTime rowInputRealizer := | |
| polynomial_pair (polynomial_pair | |
| (polynomial_composition (polynomial_composition | |
| (polynomial_composition polynomial_first polynomial_first) polynomial_first) polynomial_first) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_first)) | |
| polynomial_second | |
| lemma polynomial_columnInput : PolynomialTime columnInputRealizer := | |
| polynomial_pair (polynomial_pair | |
| (polynomial_composition (polynomial_composition | |
| (polynomial_composition polynomial_first polynomial_first) polynomial_first) polynomial_second) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_second)) | |
| polynomial_second | |
| lemma polynomial_capacityInput : PolynomialTime capacityInputRealizer := | |
| polynomial_pair (polynomial_pair | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_second) | |
| (polynomial_composition polynomial_first polynomial_second)) polynomial_second | |
| def shiftedRealizer : Realizer shifted := | |
| pair (pair | |
| (composition (f := fun z : Input=>((z.1.1.1.1,z.1.2.1),z.2)) | |
| (g := ListSubtractProgram.subtractAt) rowInputRealizer ListSubtractProgram.subtractAtRealizer) | |
| (composition (f := fun z : Input=>((z.1.1.1.2,z.1.2.2),z.2)) | |
| (g := ListSubtractProgram.subtractAt) columnInputRealizer ListSubtractProgram.subtractAtRealizer)) | |
| (composition (f := fun z : Input=>((z.1.1.2,z.1.2),z.2)) | |
| (g := LowerCapacityProgram.matrix) capacityInputRealizer LowerCapacityProgram.matrixRealizer) | |
| lemma polynomial_shifted : PolynomialTime shiftedRealizer := | |
| polynomial_pair (polynomial_pair | |
| (polynomial_composition polynomial_rowInput ListSubtractProgram.polynomial_subtractAt) | |
| (polynomial_composition polynomial_columnInput ListSubtractProgram.polynomial_subtractAt)) | |
| (polynomial_composition polynomial_capacityInput LowerCapacityProgram.polynomial_matrix) | |
| def guards (z : Input) : Bool := | |
| decide (z.2≤z.1.1.1.1.getD z.1.2.1 0) && | |
| (decide (z.2≤z.1.1.1.2.getD z.1.2.2 0) && | |
| decide (z.2≤ResidualGraphProgram.capacityEntry (z.1.1.2,z.1.2))) | |
| def guardsRealizer : Realizer guards := by | |
| let rowBound : Realizer (fun z : Input=>z.1.1.1.1.getD z.1.2.1 0) := | |
| composition (composition rowInputRealizer first) (getDFast 0) | |
| let columnBound : Realizer (fun z : Input=>z.1.1.1.2.getD z.1.2.2 0) := | |
| composition (composition columnInputRealizer first) (getDFast 0) | |
| let capacityBound : Realizer (fun z : Input=>ResidualGraphProgram.capacityEntry (z.1.1.2,z.1.2)) := | |
| composition (composition capacityInputRealizer first) ResidualGraphProgram.capacityEntryRealizer | |
| exact composition (pair (composition (pair second rowBound) leNat) | |
| (composition (pair (composition (pair second columnBound) leNat) | |
| (composition (pair second capacityBound) leNat)) boolAnd)) boolAnd | |
| lemma polynomial_guards : PolynomialTime guardsRealizer := by | |
| unfold guardsRealizer | |
| apply polynomial_composition | |
| · apply polynomial_pair | |
| · exact polynomial_composition (polynomial_pair polynomial_second | |
| (polynomial_composition (polynomial_composition polynomial_rowInput polynomial_first) | |
| (polynomial_getDFast 0))) polynomial_leNat | |
| · apply polynomial_composition | |
| · apply polynomial_pair | |
| · exact polynomial_composition (polynomial_pair polynomial_second | |
| (polynomial_composition (polynomial_composition polynomial_columnInput polynomial_first) | |
| (polynomial_getDFast 0))) polynomial_leNat | |
| · exact polynomial_composition (polynomial_pair polynomial_second | |
| (polynomial_composition (polynomial_composition polynomial_capacityInput polynomial_first) | |
| ResidualGraphProgram.polynomial_capacityEntry)) polynomial_leNat | |
| · exact polynomial_boolAnd | |
| · exact polynomial_boolAnd | |
| def test (z : Input) : Bool := guards z && NetworkFeasibilityProgram.test (shifted z) | |
| def testRealizer : Realizer test := | |
| composition (pair guardsRealizer | |
| (composition (f := shifted) (g := NetworkFeasibilityProgram.test) | |
| shiftedRealizer NetworkFeasibilityProgram.testRealizer)) boolAnd | |
| lemma polynomial_test : PolynomialTime testRealizer := | |
| polynomial_composition (polynomial_pair polynomial_guards | |
| (polynomial_composition polynomial_shifted NetworkFeasibilityProgram.polynomial_test)) | |
| polynomial_boolAnd | |
| lemma shifted_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) (t : ℕ) : | |
| shifted ((NetworkVertexCode.data r c b,(i.val,j.val)),t)= | |
| NetworkVertexCode.data (BoundedCellShift.margin r i t) | |
| (BoundedCellShift.margin c j t) (BoundedCellShift.lower b i j t) := by | |
| unfold shifted NetworkVertexCode.data | |
| rw [ListSubtractProgram.subtractAt_ofFn,ListSubtractProgram.subtractAt_ofFn, | |
| LowerCapacityProgram.matrix_code] | |
| lemma guards_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) (t : ℕ) : | |
| guards ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔ | |
| t≤r i ∧ t≤c j ∧ t≤b i j := by | |
| dsimp only [guards, NetworkVertexCode.data] | |
| rw [ResidualGraphProgram.capacityEntry_code] | |
| simp only [List.getD_eq_getElem?_getD,List.getElem?_ofFn, | |
| i.isLt,j.isLt,dite_true,Option.getD_some,Fin.eta,Bool.and_eq_true] | |
| constructor | |
| · rintro ⟨hr,hc,hb⟩ | |
| exact ⟨_root_.of_decide_eq_true hr,_root_.of_decide_eq_true hc,_root_.of_decide_eq_true hb⟩ | |
| · rintro ⟨hr,hc,hb⟩ | |
| exact ⟨_root_.decide_eq_true hr,_root_.decide_eq_true hc,_root_.decide_eq_true hb⟩ | |
| theorem test_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) (t : ℕ) : | |
| test ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔ | |
| ∃ X,HasMargins X r c ∧ (∀ u v,X u v≤b u v) ∧ t≤X i j := by | |
| rw [test,Bool.and_eq_true,guards_code,shifted_code,NetworkFeasibilityProgram.test_correct hEK] | |
| simpa only [and_assoc] using (BoundedCellShift.feasible_lower_iff r c b i j t).symm | |
| end ContingencyTables.CellLowerTestProgram | |
| end OAI | |