import OAI.Combinatorics.ContingencyTables.Dense.DenseAxisProgram

namespace OAI

/-! # Natural coordinate encodings of dense axis moves -/

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