openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Rooted /LargeMarginReference.lean
| import OAI.Combinatorics.ContingencyTables.Rooted.LargeMarginSemantics | |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseCanonicalSemantics | |
| namespace OAI | |
| /-! # Computed reference coordinates for the large completion rectangle -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms SmallContextCoordinates SmallContextFibres | |
| open PhysicalCompletionFibres FirstPaperProfiles PaddedCompletions CompletionCounts | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n q : ℕ} | |
| def optionIndexEquiv (k : ℕ) (hk : 0 < k) : Option (Fin (k-1)) ≃ Fin k := by | |
| cases k with | |
| | zero => omega | |
| | succ k => exact (finSuccEquiv k).symm | |
| lemma optionIndex_marginCode (k : ℕ) (hk : 0 < k) (f : Fin k → ℕ) : | |
| marginCode (fun i => f (optionIndexEquiv k hk i))=List.ofFn f := by | |
| cases k with | |
| | zero => omega | |
| | succ k => | |
| rw [marginCode_ofFn] | |
| apply congrArg List.ofFn | |
| funext i | |
| exact congrArg f ((finSuccEquiv k).symm_apply_apply i) | |
| def largeOptionEquiv (r : Fin m → ℕ) (U : ℕ) | |
| (hr : 0 < (largeIndices (U,List.ofFn r)).length) : | |
| Option (Fin ((largeIndices (U,List.ofFn r)).length-1)) ≃ LargeRows r U := | |
| (optionIndexEquiv _ hr).trans (largeIndexEquiv r U) | |
| lemma largeIndices_pos (r : Fin m → ℕ) (U : ℕ) (i : LargeRows r U) : | |
| 0 < (largeIndices (U,List.ofFn r)).length := by | |
| apply List.length_pos_iff.mpr | |
| intro he | |
| have hm := (largeIndices_ofFn_mem r U i.val).mpr i.property | |
| simp only [he,List.not_mem_nil] at hm | |
| lemma largeOption_marginCode (r : Fin m → ℕ) (U : ℕ) | |
| (hr : 0 < (largeIndices (U,List.ofFn r)).length) (f : LargeRows r U → ℕ) : | |
| marginCode (fun i => f (largeOptionEquiv r U hr i))= | |
| List.ofFn (fun i => f (largeIndexEquiv r U i)) := | |
| optionIndex_marginCode _ hr (fun i => f (largeIndexEquiv r U i)) | |
| def listedRows (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ) | |
| (B : Fin m → Fin n → ℕ) (hr : 0 < (largeIndices (U,List.ofFn r)).length) | |
| (z : PhysicalStates (firstPaperSmall r c U) B r c L) : | |
| Option (Fin ((largeIndices (U,List.ofFn r)).length-1)) → ℕ := | |
| fun i => residualRows (firstPaperSmall r c U) B r c L z (largeOptionEquiv r U hr i).val | |
| def listedColumns (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ) | |
| (B : Fin m → Fin n → ℕ) (hc : 0 < (largeIndices (U,List.ofFn c)).length) | |
| (z : PhysicalStates (firstPaperSmall r c U) B r c L) : | |
| Option (Fin ((largeIndices (U,List.ofFn c)).length-1)) → ℕ := | |
| fun j => residualColumns (firstPaperSmall r c U) B r c L z (largeOptionEquiv c U hc j).val | |
| theorem listedRows_code (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ) | |
| (B : Fin m → Fin n → ℕ) (hr : 0 < (largeIndices (U,List.ofFn r)).length) | |
| (e : Fin q ≃ Cells (firstPaperSmall r c U)) | |
| (z : PhysicalStates (firstPaperSmall r c U) B r c L) : | |
| marginCode (listedRows r c U L B hr z)=largePaddedMargins true | |
| (((((cellCatalog (firstPaperSmall r c U) e, | |
| viewList (firstPaperSmall r c U) e (physicalProfile _ B r c L z) false), | |
| (List.ofFn r,List.ofFn c)),U),L)) := by | |
| unfold listedRows | |
| rw [largeOption_marginCode r U hr | |
| (fun i : LargeRows r U => residualRows (firstPaperSmall r c U) B r c L z i.val), | |
| largePaddedRows_code] | |
| rfl | |
| theorem listedColumns_code (r : Fin m → ℕ) (c : Fin n → ℕ) (U L : ℕ) | |
| (B : Fin m → Fin n → ℕ) (hc : 0 < (largeIndices (U,List.ofFn c)).length) | |
| (e : Fin q ≃ Cells (firstPaperSmall r c U)) | |
| (hB : ∀ a : Cells (firstPaperSmall r c U),B a.val.1 a.val.2=U) | |
| (z : PhysicalStates (firstPaperSmall r c U) B r c L) : | |
| marginCode (listedColumns r c U L B hc z)=largePaddedMargins false | |
| (((((cellCatalog (firstPaperSmall r c U) e, | |
| complementValues (U,viewList (firstPaperSmall r c U) e (physicalProfile _ B r c L z) true)), | |
| (List.ofFn r,List.ofFn c)),U),L)) := by | |
| unfold listedColumns | |
| rw [largeOption_marginCode c U hc | |
| (fun j : LargeColumns c U => residualColumns (firstPaperSmall r c U) B r c L z j.val), | |
| largePaddedColumns_code r c U L e B hB] | |
| rfl | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |