| import OAI.Combinatorics.ContingencyTables.Dense.DenseAxisProgram |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace ContingencyTables.DensePrograms |
| open ProfilePrograms GridBoundary |
| open scoped Classical |
|
|
| lemma listExchange_increment {q : ℕ} (v : Fin q → ℕ) (i : Fin q) : |
| listExchange (List.ofFn v,(i.val,q))= |
| List.ofFn (Function.update v i (v i+1)) := by |
| apply List.ext_getElem |
| · simp only [listExchange,List.length_map,List.length_range,List.length_ofFn] |
| · intro j hj _ |
| have hjq : j<q := by |
| simpa only [listExchange,List.length_map,List.length_range,List.length_ofFn] using hj |
| simp only [listExchange,List.getElem_map,List.getElem_range,List.getElem_ofFn, |
| exchangeEntry,List.getD_eq_getElem?_getD,List.getElem?_ofFn,hjq,dite_true, |
| Option.getD_some,ite_eq_right (Nat.ne_of_lt hjq),Nat.sub_zero] |
| by_cases hji : (⟨j,hjq⟩ : Fin q)=i |
| · subst i |
| simp only [Function.update_self,ite_true] |
| · have hne : j≠i.val := fun h => hji (Fin.ext h) |
| simp only [hne,ite_false,Nat.add_zero,Function.update_of_ne hji] |
|
|
| lemma listExchange_decrement {q : ℕ} (v : Fin q → ℕ) (i : Fin q) : |
| listExchange (List.ofFn v,(q,i.val))= |
| List.ofFn (Function.update v i (v i-1)) := by |
| apply List.ext_getElem |
| · simp only [listExchange,List.length_map,List.length_range,List.length_ofFn] |
| · intro j hj _ |
| have hjq : j<q := by |
| simpa only [listExchange,List.length_map,List.length_range,List.length_ofFn] using hj |
| simp only [listExchange,List.getElem_map,List.getElem_range,List.getElem_ofFn, |
| exchangeEntry,List.getD_eq_getElem?_getD,List.getElem?_ofFn,hjq,dite_true, |
| Option.getD_some,ite_eq_right (Nat.ne_of_lt hjq),Nat.add_zero] |
| by_cases hji : (⟨j,hjq⟩ : Fin q)=i |
| · subst i |
| simp only [Function.update_self,ite_true] |
| · have hne : j≠i.val := fun h => hji (Fin.ext h) |
| simp only [hne,ite_false,Nat.sub_zero,Function.update_of_ne hji] |
|
|
| def gridNatCode {q K : ℕ} (a : Grid (I := Fin q) K) : List ℕ := |
| List.ofFn (fun i => (corner a i).toNat) |
|
|
| lemma gridNatCode_getD {q K : ℕ} (a : Grid (I := Fin q) K) (i : Fin q) : |
| (gridNatCode a).getD i.val 0=(corner a i).toNat := by |
| simp only [gridNatCode,List.getD_eq_getElem?_getD,List.getElem?_ofFn,i.isLt, |
| dite_true,Option.getD_some,Fin.eta] |
|
|
| lemma gridNatCode_update {q K : ℕ} (a : Grid (I := Fin q) K) (i : Fin q) |
| (v : Set.Icc (0 : ℤ) ((K : ℤ)-1)) : |
| gridNatCode (Function.update a i v)= |
| List.ofFn (Function.update (fun j => (corner a j).toNat) i v.val.toNat) := by |
| apply congrArg List.ofFn |
| funext j |
| by_cases hji : j=i |
| · subst j |
| simp only [Function.update_self,corner] |
| · simp only [Function.update_of_ne hji,corner] |
|
|
| /-- The bounded list operation implements exactly the signed geometric |
| axis proposal, including both boundary rejections. -/ |
| theorem axisListStep_eq {q K : ℕ} (a : Grid (I := Fin q) K) (i : Fin q) (sign : Bool) : |
| axisListStep (((K,gridNatCode a),i.val),sign)= |
| (axisStep a i sign).map gridNatCode := by |
| have hlen : (gridNatCode a).length=q := by simp only [gridNatCode,List.length_ofFn] |
| have hnonneg : 0≤corner a i := (a i).property.1 |
| have hupper : corner a i≤(K : ℤ)-1 := (a i).property.2 |
| have hcast : ((corner a i).toNat : ℤ)=corner a i := Int.toNat_of_nonneg hnonneg |
| cases sign with |
| | false => |
| simp only [axisListStep,hlen,i.isLt,ite_true,Bool.false_eq_true,ite_false, |
| gridNatCode_getD,axisStep] |
| by_cases h : 0≤corner a i-1 |
| · have hn : 0<(corner a i).toNat := by omega |
| rw [ite_eq_left hn,dite_eq_left h,Option.map_some] |
| let v : Set.Icc (0 : ℤ) ((K : ℤ)-1) := ⟨corner a i-1,h,by omega⟩ |
| have hc := gridNatCode_update a i v |
| have he : v.val.toNat=(corner a i).toNat-1 := by dsimp only [v]; omega |
| rw [he] at hc |
| exact congrArg some ((listExchange_decrement (fun j => (corner a j).toNat) i).trans hc.symm) |
| · have hn : ¬0<(corner a i).toNat := by omega |
| rw [ite_eq_right hn,dite_eq_right h,Option.map_none] |
| | true => |
| simp only [axisListStep,hlen,i.isLt,ite_true,gridNatCode_getD,axisStep] |
| by_cases h : corner a i+1≤(K : ℤ)-1 |
| · have hn : (corner a i).toNat+1<K := by omega |
| rw [ite_eq_left hn,dite_eq_left h,Option.map_some] |
| let v : Set.Icc (0 : ℤ) ((K : ℤ)-1) := ⟨corner a i+1,by omega,h⟩ |
| have hc := gridNatCode_update a i v |
| have he : v.val.toNat=(corner a i).toNat+1 := by dsimp only [v]; omega |
| rw [he] at hc |
| exact congrArg some ((listExchange_increment (fun j => (corner a j).toNat) i).trans hc.symm) |
| · have hn : ¬(corner a i).toNat+1<K := by omega |
| rw [ite_eq_right hn,dite_eq_right h,Option.map_none] |
|
|
| end ContingencyTables.DensePrograms |
|
|
| end OAI |
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|
