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