import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitializedRestartSemantics
import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartNativeLaw
import OAI.Combinatorics.ContingencyTables.Machines.FiniteEncodedBind
import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartPadding

namespace OAI

/-! # Exact law of the complete executed initializer and restart prefix -/
namespace ContingencyTables.CompanionInitializedRestartProgram
open MatchingFPRAS CommonBasesFPRAS FirstSuccess FairBits FiniteObservableTransport
open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal
open CompanionArithmeticProposal CompanionEvaluatedWeightProgram
open CompanionOracleDensity TwoLiftStates GridBoundary
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ}

def parameters (A : Instance (Fin m) (Fin n)) (H T N R L : ℕ)
    (epsilon : TreeFraction.Fraction) : CountingPhaseParametersProgram.Input :=
  ([Fintype.card (Labels A),A.W,H,T,N,R,L],epsilon)

def model (A : Instance (Fin m) (Fin n)) (C K H T N R L j : ℕ)
    (epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
    (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
    (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction)
    (bits : Fin (CompanionInitialDraw.initialBits A
        (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
      CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) → Bool) :
    PlainIndex (pairs : List (CompanionRationalUpdates.Pair A)) :=
  let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)
  let widths := CompanionRestartProgram.sourceWidth A C K (parameters A H T N R L epsilon)
  let split := splitWordEquiv (CompanionInitialDraw.initialBits A B)
    (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) bits
  varyingWalk widths
    (CompanionRestartProgram.sourceStep A C K (parameters A H T N R L epsilon) hK hW v w)
    (CompanionInitialDraw.draw A B split.1) j (varyingWordEquiv widths j split.2)

 /-- The actual list program has the complete finite-word law, including
 every initialization bias, rejected return and fallback. -/
theorem output_padded_sameLaw (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ)
    (hC : 0 < C) (hj : j ≤ rationalHeight A C) (hK : 1 ≤ K)
    (hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=CompanionOracleDensity.binWidth A C)
    (epsilon : TreeFraction.Fraction)
    (hxi : 0 < TreeFraction.val (CountingPhaseParametersProgram.noise (parameters A H T N R L epsilon)))
    (z : View A → ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
    (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction)
    (estimates : Fin j → TreeFraction.Fraction)
    (hdegree : ∀ i : Fin j,RootedOracleConfigProgram.rootDegree
      (CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v
        (List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
          (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤
      wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
        (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding))
    (hdepth : ∀ i : Fin j,RootedOracleConfigProgram.depth
      (CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v
        (List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
          (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤
      wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
        (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) :
    SameLaw (fun bits : Fin (wordLength A
        (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
      (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) → Bool =>
      output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)))
      (fun bits => encodeSet (coordinateOrder A) (model A C K H T N R L j epsilon hK hW v w bits).val.val) := by
  let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)
  let p := parameters A H T N R L epsilon
  let n := CompanionRestartProgram.sumBits A C K j p
  let reserve := List.replicate (wordLength A B+(n+padding)) false
  let stageList := CompanionRestartProgram.stagesData A C K j p z v reserve w
  let init := fun bits : Fin (wordLength A B) → Bool => CompanionRestartInitializer.output
    (CompanionRestartInitializer.data A C K H T N R L epsilon z v,List.ofFn bits)
  let realInit := CompanionInitialDraw.draw A B
  let encode := fun S : PlainIndex (pairs : List (CompanionRationalUpdates.Pair A)) =>
    encodeSet (coordinateOrder A) S.val.val
  let run := fun code (bits : Fin (n+padding) → Bool) =>
    CompanionRestartProgram.output ((stageList,code),List.ofFn bits)
  let realRun := fun S (bits : Fin n → Bool) => encode (varyingWalk
    (CompanionRestartProgram.sourceWidth A C K p)
    (CompanionRestartProgram.sourceStep A C K p hK hW v w) S j
    (varyingWordEquiv (CompanionRestartProgram.sourceWidth A C K p) j bits))
  have hi : SameLaw init (fun bits => encode (realInit bits)) :=
    CompanionRestartInitializer.output_sameLaw A C K H T N R L epsilon z v
  have hs : ∀ S,SameLaw (run (encode S)) (realRun S) := by
    intro S
    have hn := CompanionRestartProgram.widthSum_data A C K j p z v reserve w S.val.val
    have hh := CompanionRestartProgram.output_native_sameLaw A C K j hC hj hK hW hB p hxi z v reserve
      (fun i => by simpa only [reserve,List.length_replicate] using hdegree i)
      (fun i => by simpa only [reserve,List.length_replicate] using hdepth i) w S
    exact restart_append_sameLaw (CompanionRestartProgram.baseData A C K j p z v reserve w S.val.val)
      n hn (realRun S) hh padding
  have hf := flat_encoded_bind (wordLength A B) (n+padding) (CompanionInitialDraw.initialBits A B) n
    init realInit encode run realRun hi hs
  intro phi
  have h := hf phi
  have he : (fun bits : Fin (wordLength A B+(n+padding)) → Bool =>
      output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)))=
      (fun bits => run (init (splitWordEquiv (wordLength A B) (n+padding) bits).1)
        (splitWordEquiv (wordLength A B) (n+padding) bits).2) := by
    funext bits
    exact output_split A C K H T N R L j (n+padding) epsilon z v w estimates bits
  rw [he]
  exact h

 /-- When the supplied restart time is the computed paper schedule,
 the reference experiment is precisely initializedStages. -/
theorem model_eq_initializedStages (A : Instance (Fin m) (Fin n)) (C K H N R L j : ℕ)
    (epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
    (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
    (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction)
    (bits : Fin (RootedRestartWords.initialStageBits A C K H L
      (CountingPhaseParametersProgram.precision
        (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon))
      R (CountingPhaseSchedule.initialPrecision H) j
      (TreeFraction.val (CountingPhaseParametersProgram.noise
        (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon)))) → Bool) :
    model A C K H (CompanionTargetMixing.restartTime A H L) N R L j epsilon hK hW v w bits=
      RootedRestartWords.initializedStages A (RootedCompletion.rowColumn A.T) C K H L
        (CountingPhaseParametersProgram.precision
          (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon))
        R (CountingPhaseSchedule.initialPrecision H) j hK hW
        (TreeFraction.val (CountingPhaseParametersProgram.noise
          (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon)))
        (RootedCatalogGrid.gridEquiv A K v) (fun i => sourceWeights A (w i)) bits := rfl

end
end ContingencyTables.CompanionInitializedRestartProgram

end OAI