import OAI.Combinatorics.ContingencyTables.Transport.BoundedProfileWalk
import OAI.Combinatorics.ContingencyTables.Transport.ListedBooleanStep

namespace OAI

/-! # The bounded binary loop preserves the physical profile walk -/

namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms PhysicalCompletionFibres FirstPaperPhysicalMarginal CompletionCounts
open SmallContextCoordinates FirstSuccess FairBits
open scoped BigOperators Classical

noncomputable section
variable {m n q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
variable (e : Fin q ≃ Cells (paperSmall r c))
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)

lemma stateCode_bounded (x : PaperStates r c) (a : Cells (paperSmall r c)) (side : Bool) :
    physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) x (a,side)≤d^20 := by
  have he := (physicalProfile_feasible r c x).bounded (a,side)
  exact he.trans_eq (stateCapacity_small r c _ _ a.property)

lemma capViews_stateCode (x y : PaperStates r c) :
    capViews (stateCodeData r c e x,stateCode r c e y)=stateCode r c e y := by
  have hb (side : Bool) :
      capView ((q,d^20),viewList (paperSmall r c) e
        (physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) y) side)=
      viewList (paperSmall r c) e
        (physicalProfile (paperSmall r c) (stateCapacity r c) r c (d^12) y) side := by
    apply capView_eq
    · simp only [viewList,List.length_ofFn]
    · intro a ha
      obtain ⟨i,rfl⟩ := List.mem_ofFn.mp ha
      exact stateCode_bounded r c y (e i) side
  simpa only [capViews,stateCodeData,codeCheckData,cellCatalog,List.length_ofFn,stateCode,profileCode]
    using Prod.ext (hb false) (hb true)

variable (hr : 0 < (largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn r)).length)
variable (hc : 0 < (largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn c)).length)
variable (b h : ℕ) (hb : Nat.clog 2 (stateBound r c+2) ≤ b)
local notation "s" => smallProposalBits (I := Fin m) (J := Fin n)
local notation "f" => 25*(d+b+h+1)^62

def encodedConfiguration (initial current : PaperStates r c) : ProfileConfiguration :=
  (((stateCodeData r c e initial,d^12),(d,h)),stateCode r c e current)

theorem advanceProfile_encoded (htotal : (∑ i,r i)=∑ j,c j)
    (initial x : PaperStates r c) (bits : Fin (s+f) → Bool) :
    advanceProfile ((List.ofFn (splitWordEquiv s f bits).1,List.ofFn (splitWordEquiv s f bits).2),
      encodedConfiguration r c e h initial x)=
      encodedConfiguration r c e h initial (listedBitStep r c e hr hc b h hb x bits) := by
  apply Prod.ext
  · rfl
  · change capViews (stateCodeData r c e initial,
      profileStep (encodedStepInput r c e b h x
        (splitWordEquiv s f bits).1 (splitWordEquiv s f bits).2))=_
    rw [profileStep_listedBitStep r c e hr hc b h hb htotal]
    exact capViews_stateCode r c e initial _

theorem profileWalk_ofFn (htotal : (∑ i,r i)=∑ j,c j)
    (initial x : PaperStates r c) (steps : ℕ) (bits : Fin steps → Fin (s+f) → Bool) :
    profileWalk (encodedConfiguration r c e h initial x,
      List.ofFn (fun i => (List.ofFn (splitWordEquiv s f (bits i)).1,
        List.ofFn (splitWordEquiv s f (bits i)).2)))=
      encodedConfiguration r c e h initial
        (wordWalk (listedBitStep r c e hr hc b h hb) x steps bits) := by
  induction steps with
  | zero => rfl
  | succ steps ih =>
    unfold profileWalk at ih ⊢
    rw [List.ofFn_succ',List.concat_eq_append,List.foldl_append]
    change advanceProfile ((List.ofFn (splitWordEquiv s f (bits (Fin.last steps))).1,
      List.ofFn (splitWordEquiv s f (bits (Fin.last steps))).2),
      (List.ofFn (fun i => (List.ofFn (splitWordEquiv s f ((Fin.init bits) i)).1,
        List.ofFn (splitWordEquiv s f ((Fin.init bits) i)).2))).foldl
          (fun state word => advanceProfile (word,state))
          (encodedConfiguration r c e h initial x))=_
    rw [ih (Fin.init bits)]
    exact advanceProfile_encoded r c e hr hc b h hb htotal initial _ _

end
end ContingencyTables.SmallGraphProfiles

end OAI