Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Games.Explicit
import OAI.Computability.DirectedFeedback.Games.FinalParameters

namespace OAI

namespace DFVSGames.Reduction.ActualSource

open scoped BigOperators

structure NoiseEnumeration {s d : Nat} (g : SplitGadget s d) where
  indices : List g.NoiseIndex
  nodup : indices.Nodup
  complete : ∀ i, i ∈ indices

namespace NoiseEnumeration

variable {s d : Nat} {g : SplitGadget s d}

noncomputable def ofFintype (g : SplitGadget s d) : NoiseEnumeration g where
  indices := Finset.univ.toList
  nodup := Finset.nodup_toList _
  complete i := by simp

theorem nonempty (e : NoiseEnumeration g) : e.indices ≠ [] := by
  intro h
  obtain ⟨i⟩ := g.noiseNonempty
  have hi := e.complete i
  simp only [h, List.not_mem_nil] at hi

theorem perm_univ (e : NoiseEnumeration g) :
    e.indices.Perm (Finset.univ : Finset g.NoiseIndex).toList := by
  apply (List.perm_ext_iff_of_nodup e.nodup (Finset.nodup_toList _)).2
  intro i
  simp [e.complete i]

theorem length_eq_card (e : NoiseEnumeration g) :
    e.indices.length = Fintype.card g.NoiseIndex := by
  simpa using e.perm_univ.length_eq

theorem sum_eq_univ (e : NoiseEnumeration g) (f : g.NoiseIndex → ℚ) :
    (e.indices.map f).sum = ∑ i, f i := by
  rw [(e.perm_univ.map f).sum_eq, Finset.sum_map_toList]

theorem average_eq_expect (e : NoiseEnumeration g) (f : g.NoiseIndex → ℚ) :
    (e.indices.map f).sum / e.indices.length = Finset.univ.expect f := by
  rw [e.sum_eq_univ, e.length_eq_card, Fintype.expect_eq_sum_div_card]

end NoiseEnumeration
end DFVSGames.Reduction.ActualSource

namespace DFVSGames.Reduction.ActualGame

open DFVSGames.Integration.BinaryLinear
open ActualSource
open Foundations.Target
open scoped BigOperators

def names (S : Source) (i : Fin S.occurrences) (j : Fin 3) : Fin S.«variables» :=
  if j = 0 then (S.equation i).first
  else if j = 1 then (S.equation i).second else (S.equation i).third

def rhs (S : Source) (i : Fin S.occurrences) : F2 := ofBit (S.equation i).rhs

abbrev Question (S : Source) (k : Nat) := Fin k → Fin S.occurrences
abbrev Map (k s d : Nat) := ActualHomogeneous.E k →ₗ[F2] ActualSource.Ambient s d
abbrev Dual (k : Nat) := ActualHomogeneous.E k →ₗ[F2] F2
abbrev Query (S : Source) (k s d : Nat) := Question S k × Map k s d

noncomputable instance mapFintype (k s d : Nat) : Fintype (Map k s d) :=
  Fintype.ofInjective (fun X : Map k s d => (X : ActualHomogeneous.E k →
    ActualSource.Ambient s d)) DFunLike.coe_injective

noncomputable instance dualFintype (k : Nat) : Fintype (Dual k) :=
  Fintype.ofInjective (fun X : Dual k => (X : ActualHomogeneous.E k → F2))
    DFunLike.coe_injective

def canonical (S : Source) (k s d : Nat) (q : Query S k s d) :=
  ActualCanonical.canonical q.1 (names S) (rhs S) q.2

abbrev Vertex (S : Source) (k s d : Nat) := ActualOrbit.Vertex (canonical S k s d)
abbrev TwoSidedVertex (S : Source) (k s d : Nat) := Bool × Vertex S k s d

def vertex (S : Source) (k s d : Nat) (q : Query S k s d) : Vertex S k s d :=
  ActualOrbit.vertex (canonical S k s d) q

def offset (S : Source) (k s d : Nat) (q : Query S k s d) : Alphabet s :=
  ActualOrbit.offset (canonical S k s d) q

def shiftQuery (S : Source) (k s d : Nat) (c : Alphabet s)
    (q : Query S k s d) : Query S k s d :=
  (q.1, q.2 + ActualHomogeneous.tau.smulRight (c, (0 : Vector d)))

theorem canonical_shiftQuery (S : Source) (k s d : Nat) (c : Alphabet s)
    (q : Query S k s d) :
    canonical S k s d (shiftQuery S k s d c q) =
      ((canonical S k s d q).1 + (c, 0), (canonical S k s d q).2) :=
  ActualCanonical.canonical_shift _ _ _ _ _

@[simp] theorem vertex_shiftQuery (S : Source) (k s d : Nat) (c : Alphabet s)
    (q : Query S k s d) :
    vertex S k s d (shiftQuery S k s d c q) = vertex S k s d q := by
  apply (ActualOrbit.vertex_eq_iff _ _ _).2
  simp only [ActualOrbit.body, canonical_shiftQuery, Prod.snd_add, add_zero]

@[simp] theorem offset_shiftQuery (S : Source) (k s d : Nat) (c : Alphabet s)
    (q : Query S k s d) :
    offset S k s d (shiftQuery S k s d c q) = offset S k s d q + c := by
  simp only [offset, ActualOrbit.offset, canonical_shiftQuery, Prod.fst_add]

noncomputable def vertexCount (S : Source) (k s d : Nat) : Nat :=
  Fintype.card (TwoSidedVertex S k s d)

noncomputable def vertexEncoding (S : Source) (k s d : Nat) :
    TwoSidedVertex S k s d ≃ Fin (vertexCount S k s d) :=
  ActualOrbit.vertexEquiv (canonical S k s d)

theorem vertexCount_eq_twice (S : Source) (k s d : Nat) :
    vertexCount S k s d = 2 * Fintype.card (Vertex S k s d) := by
  simp [vertexCount, TwoSidedVertex]

abbrev Outcome (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :=
  Query S k s d × g.NoiseIndex × Dual k

def leftQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Query S k s d := ω.1

def rightQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Query S k s d :=
  (ω.1.1, ω.1.2 + ω.2.2.smulRight (g.noise ω.2.1))

noncomputable def edge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Constraint (vertexCount S k s d) (2^s) where
  source := vertexEncoding S k s d (false, vertex S k s d (leftQuery S k g ω))
  target := vertexEncoding S k s d (true, vertex S k s d (rightQuery S k g ω))
  permutation := Encoding.translationTable
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))

theorem edge_source_ne_target (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : Outcome S k g) :
    (edge S k g ω).source ≠ (edge S k g ω).target := by
  intro h
  have h' := (vertexEncoding S k s d).injective h
  have hside : false = true := congrArg Prod.fst h'
  exact Bool.false_ne_true hside

noncomputable def outcomes (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :
    List (Outcome S k g) := Finset.univ.toList

@[simp] theorem mem_outcomes (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : Outcome S k g) : ω ∈ outcomes S k g := by
  simp [outcomes]

theorem outcomes_nodup (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :
    (outcomes S k g).Nodup := Finset.nodup_toList _

@[simp] theorem outcomes_length (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) : (outcomes S k g).length = Fintype.card (Outcome S k g) := by
  simp [outcomes]

noncomputable def outputInstance (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :
    Foundations.Target.Instance (2^s) where
  vertices := vertexCount S k s d
  constraints := (outcomes S k g).map (edge S k g)
  nonempty := by
    intro h
    let ω : Outcome S k g := Classical.choice inferInstance
    have hm : edge S k g ω ∈ (outcomes S k g).map (edge S k g) :=
      List.mem_map.mpr ⟨ω, mem_outcomes S k g ω, rfl⟩
    rw [h] at hm
    exact List.not_mem_nil hm

@[simp] theorem outputInstance_constraints_length (S : Source) (k : Nat)
    {s d : Nat} (g : SplitGadget s d) :
    (outputInstance S k g).constraints.length = Fintype.card (Outcome S k g) := by
  simp [outputInstance]

noncomputable def sideLabel (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (side : Bool) :
    Vertex S k s d → Alphabet s := fun v =>
  (Encoding.alphabetEquiv s).symm (labeling (vertexEncoding S k s d (side, v)))

noncomputable def labelingOf (S : Source) (k s d : Nat)
    (labels : Bool → Vertex S k s d → Alphabet s) :
    Fin (vertexCount S k s d) → Fin (2^s) := fun i =>
  let v := (vertexEncoding S k s d).symm i
  Encoding.alphabetEquiv s (labels v.1 v.2)

@[simp] theorem sideLabel_labelingOf (S : Source) (k s d : Nat)
    (labels : Bool → Vertex S k s d → Alphabet s) (side : Bool) :
    sideLabel S k s d (labelingOf S k s d labels) side = labels side := by
  funext v
  simp [sideLabel, labelingOf]

noncomputable def unfolded (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (side : Bool)
    (q : Query S k s d) : Alphabet s :=
  ActualOrbit.unfold (canonical S k s d) (sideLabel S k s d labeling side) q

theorem unfolded_equivariant (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (side : Bool)
    (q : Query S k s d) (c : Alphabet s) :
    unfolded S k s d labeling side (shiftQuery S k s d c q) =
      unfolded S k s d labeling side q + c := by
  change sideLabel S k s d labeling side (vertex S k s d (shiftQuery S k s d c q)) +
    offset S k s d (shiftQuery S k s d c q) = _
  rw [vertex_shiftQuery, offset_shiftQuery]
  exact (add_assoc _ _ _).symm

theorem edge_satisfied_iff (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (ω : Outcome S k g) :
    (edge S k g ω).satisfied labeling = true ↔
      unfolded S k s d labeling false (leftQuery S k g ω) =
      unfolded S k s d labeling true (rightQuery S k g ω) := by
  rw [Constraint.satisfied, decide_eq_true_eq]
  have h := Encoding.translationTable_satisfied_iff
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))
    (sideLabel S k s d labeling false (vertex S k s d (leftQuery S k g ω)))
    (sideLabel S k s d labeling true (vertex S k s d (rightQuery S k g ω)))
  simp only [sideLabel, Encoding.alphabetEquiv_apply_symm_apply] at h
  exact h.trans (ActualOrbit.constraint_iff_unfolded (canonical S k s d)
    (sideLabel S k s d labeling false) (sideLabel S k s d labeling true) _ _)

noncomputable def acceptanceProbability (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : ℚ :=
  Finset.univ.expect (fun ω : Outcome S k g =>
    if (edge S k g ω).satisfied labeling then (1 : ℚ) else 0)

theorem acceptanceProbability_eq_test (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    acceptanceProbability S k g labeling =
      Finset.univ.expect (fun ω : Outcome S k g =>
        if unfolded S k s d labeling false (leftQuery S k g ω) =
            unfolded S k s d labeling true (rightQuery S k g ω)
        then (1 : ℚ) else 0) := by
  unfold acceptanceProbability
  apply Finset.expect_congr rfl
  intro ω _
  simp only [edge_satisfied_iff]

private theorem countSatisfied_map_inline_ActualGame {n q : Nat} {I : Type*}
    (labeling : Fin n → Fin q) (edges : I → Constraint n q) (xs : List I) :
    countSatisfied labeling (xs.map edges) =
      (xs.map (fun x => if (edges x).satisfied labeling then 1 else 0)).sum := by
  induction xs with
  | nil => rfl
  | cons x xs ih => simp only [List.map_cons, countSatisfied, List.sum_cons, ih]

theorem countSatisfied_eq_sum (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    countSatisfied labeling (outputInstance S k g).constraints =
      ∑ ω : Outcome S k g, if (edge S k g ω).satisfied labeling then 1 else 0 := by
  change countSatisfied labeling ((outcomes S k g).map (edge S k g)) = _
  rw [countSatisfied_map_inline_ActualGame]
  exact Finset.sum_map_toList _ _

theorem acceptanceProbability_eq_count (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    acceptanceProbability S k g labeling =
      (countSatisfied labeling (outputInstance S k g).constraints : ℚ) /
        (outputInstance S k g).constraints.length := by
  rw [acceptanceProbability, Fintype.expect_eq_sum_div_card,
    outputInstance_constraints_length, countSatisfied_eq_sum]
  congr 1
  simp

def orbitBodyDecidableEq (S : Source) (k s d : Nat) :
    DecidableEq (Vector d × (Fin k → ActualCanonical.Record
      (Fin S.«variables») (Fin S.occurrences) (ActualSource.Ambient s d))) :=
  @instDecidableEqProd _ _ inferInstance inferInstance

def explicitBodies (S : Source) (k s d : Nat) := by
  letI := orbitBodyDecidableEq S k s d
  exact Encoding.imageVertices (ActualEnumeration.queries S.occurrences k s d)
    (ActualOrbit.body (canonical S k s d))

def explicitVertexCount (S : Source) (k s d : Nat) : Nat :=
  2 * (explicitBodies S k s d).length

def explicitVertexEncoding (S : Source) (k s d : Nat) :
    TwoSidedVertex S k s d ≃ Fin (explicitVertexCount S k s d) := by
  letI := orbitBodyDecidableEq S k s d
  exact (Equiv.prodCongr (Equiv.refl Bool)
    (ActualOrbit.explicitVertexEquiv (canonical S k s d)
      (ActualEnumeration.queries S.occurrences k s d) ActualEnumeration.mem_queries)).trans
    (Encoding.sideEquiv _)

noncomputable def semanticToExplicitVertices (S : Source) (k s d : Nat) :
    Fin (vertexCount S k s d) ≃ Fin (explicitVertexCount S k s d) :=
  (vertexEncoding S k s d).symm.trans (explicitVertexEncoding S k s d)

@[simp] theorem semanticToExplicitVertices_apply (S : Source) (k s d : Nat)
    (v : TwoSidedVertex S k s d) :
    semanticToExplicitVertices S k s d (vertexEncoding S k s d v) =
      explicitVertexEncoding S k s d v := by
  simp [semanticToExplicitVertices]

theorem explicitVertexCount_eq (S : Source) (k s d : Nat) :
    explicitVertexCount S k s d = vertexCount S k s d := by
  simpa using (Fintype.card_congr (semanticToExplicitVertices S k s d)).symm

def explicitEdge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Constraint (explicitVertexCount S k s d) (2^s) where
  source := explicitVertexEncoding S k s d (false, vertex S k s d (leftQuery S k g ω))
  target := explicitVertexEncoding S k s d (true, vertex S k s d (rightQuery S k g ω))
  permutation := Encoding.translationTable
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))

theorem explicitEdge_eq_rename (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : Outcome S k g) :
    explicitEdge S k g ω = Integration.InstanceEquivalences.renameConstraint
      (semanticToExplicitVertices S k s d) (edge S k g ω) := by
  simp only [explicitEdge, Integration.InstanceEquivalences.renameConstraint, edge,
    semanticToExplicitVertices_apply]

def explicitOutcomes (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (en : NoiseEnumeration g) : List (Outcome S k g) :=
  ActualEnumeration.indexedOutcomes S.occurrences k s d en.indices

theorem explicitOutcomes_nodup (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (explicitOutcomes S k g en).Nodup :=
  ActualEnumeration.nodup_indexedOutcomes en.nodup _ _ _ _

@[simp] theorem mem_explicitOutcomes (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) (ω : Outcome S k g) :
    ω ∈ explicitOutcomes S k g en :=
  ActualEnumeration.mem_indexedOutcomes en.complete ω

theorem explicitOutcomes_perm_outcomes (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (explicitOutcomes S k g en).Perm (outcomes S k g) := by
  apply (List.perm_ext_iff_of_nodup (explicitOutcomes_nodup S k g en)
    (outcomes_nodup S k g)).2
  intro ω
  simp

def outputInstanceWithEnumeration (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) : Foundations.Target.Instance (2^s) where
  vertices := explicitVertexCount S k s d
  constraints := (explicitOutcomes S k g en).map (explicitEdge S k g)
  nonempty := by
    intro h
    let ω : Outcome S k g := Classical.choice inferInstance
    have hm : explicitEdge S k g ω ∈
        (explicitOutcomes S k g en).map (explicitEdge S k g) :=
      List.mem_map.mpr ⟨ω, mem_explicitOutcomes S k g en ω, rfl⟩
    rw [h] at hm
    exact List.not_mem_nil hm

@[simp] theorem outputInstanceWithEnumeration_length (S : Source) (k : Nat)
    {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (outputInstanceWithEnumeration S k g en).constraints.length =
      Explicit.edgeCount S.occurrences k (s+d) en.indices.length := by
  simp only [outputInstanceWithEnumeration, List.length_map, explicitOutcomes,
    ActualEnumeration.length_indexedOutcomes]

theorem explicitConstraints_perm_renamed (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (outputInstanceWithEnumeration S k g en).constraints.Perm
      ((outputInstance S k g).constraints.map
        (Integration.InstanceEquivalences.renameConstraint (semanticToExplicitVertices S k s d))) := by
  simpa only [outputInstanceWithEnumeration, outputInstance, List.map_map,
    Function.comp_def, ← explicitEdge_eq_rename] using
    (explicitOutcomes_perm_outcomes S k g en).map (explicitEdge S k g)

theorem completeAt_outputInstanceWithEnumeration_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    CompleteAt error (outputInstanceWithEnumeration S k g en) ↔
      CompleteAt error (outputInstance S k g) :=
  Integration.InstanceEquivalences.completeAt_iff_of_rename_perm error
    (outputInstance S k g) (outputInstanceWithEnumeration S k g en)
    (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en)

theorem soundAt_outputInstanceWithEnumeration_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    SoundAt error (outputInstanceWithEnumeration S k g en) ↔
      SoundAt error (outputInstance S k g) :=
  Integration.InstanceEquivalences.soundAt_iff_of_rename_perm error
    (outputInstance S k g) (outputInstanceWithEnumeration S k g en)
    (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en)

end DFVSGames.Reduction.ActualGame

namespace DFVSGames.Decoder.TableKeysGame

open DFVSGames.Integration.BinaryLinear DFVSGames.Reduction
open ActualSource Foundations.Target
open scoped BigOperators

abbrev Question (S : Source) (k : Nat) := ActualGame.Question S k
abbrev Map (k s d : Nat) := ActualGame.Map k s d
abbrev Dual (k : Nat) := ActualGame.Dual k
abbrev Query (S : Source) (k s d : Nat) := ActualGame.Query S k s d
abbrev canonical (S : Source) (k s d : Nat) := ActualGame.canonical S k s d

abbrev Vertex (S : Source) (k s d : Nat) := ActualGame.Vertex S k s d
abbrev vertex (S : Source) (k s d : Nat) := ActualGame.vertex S k s d
abbrev offset (S : Source) (k s d : Nat) := ActualGame.offset S k s d
abbrev shiftQuery (S : Source) (k s d : Nat) := ActualGame.shiftQuery S k s d

noncomputable def vertexCount (S : Source) (k s d : Nat) : Nat :=
  Fintype.card (Vertex S k s d)

noncomputable def vertexEncoding (S : Source) (k s d : Nat) :
    Vertex S k s d ≃ Fin (vertexCount S k s d) := Fintype.equivFin _

abbrev Outcome (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :=
  Query S k s d × g.NoiseIndex × Dual k

abbrev leftQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :=
  ActualGame.leftQuery S k g
abbrev rightQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :=
  ActualGame.rightQuery S k g

noncomputable def edge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Constraint (vertexCount S k s d) (2^s) where
  source := vertexEncoding S k s d (vertex S k s d (leftQuery S k g ω))
  target := vertexEncoding S k s d (vertex S k s d (rightQuery S k g ω))
  permutation := Encoding.translationTable
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))

noncomputable def outcomes (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :=
  ActualGame.outcomes S k g

noncomputable def outputInstance (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) :
    Instance (2^s) where
  vertices := vertexCount S k s d
  constraints := (outcomes S k g).map (edge S k g)
  nonempty := by
    intro h
    let ω : Outcome S k g := Classical.choice inferInstance
    have hm : edge S k g ω ∈ (outcomes S k g).map (edge S k g) :=
      List.mem_map.mpr ⟨ω, ActualGame.mem_outcomes S k g ω, rfl⟩
    rw [h] at hm
    exact List.not_mem_nil hm

@[simp] theorem outputInstance_constraints_length (S : Source) (k : Nat)
    {s d : Nat} (g : SplitGadget s d) :
    (outputInstance S k g).constraints.length = Fintype.card (Outcome S k g) := by
  simp [outputInstance, outcomes]

noncomputable def vertexLabel (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    Vertex S k s d → Alphabet s := fun v =>
  (Encoding.alphabetEquiv s).symm (labeling (vertexEncoding S k s d v))

noncomputable def labelingOf (S : Source) (k s d : Nat)
    (labels : Vertex S k s d → Alphabet s) :
    Fin (vertexCount S k s d) → Fin (2^s) := fun i =>
  Encoding.alphabetEquiv s (labels ((vertexEncoding S k s d).symm i))

@[simp] theorem vertexLabel_labelingOf (S : Source) (k s d : Nat)
    (labels : Vertex S k s d → Alphabet s) :
    vertexLabel S k s d (labelingOf S k s d labels) = labels := by
  funext v
  simp [vertexLabel, labelingOf]

noncomputable def unfolded (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s))
    (q : Query S k s d) : Alphabet s :=
  ActualOrbit.unfold (canonical S k s d) (vertexLabel S k s d labeling) q

theorem unfolded_equivariant (S : Source) (k s d : Nat)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s))
    (q : Query S k s d) (c : Alphabet s) :
    unfolded S k s d labeling (shiftQuery S k s d c q) =
      unfolded S k s d labeling q + c := by
  change vertexLabel S k s d labeling
      (ActualGame.vertex S k s d (ActualGame.shiftQuery S k s d c q)) +
    ActualGame.offset S k s d (ActualGame.shiftQuery S k s d c q) = _
  rw [ActualGame.vertex_shiftQuery, ActualGame.offset_shiftQuery]
  exact (add_assoc _ _ _).symm

theorem edge_satisfied_iff (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (ω : Outcome S k g) :
    (edge S k g ω).satisfied labeling = true ↔
      unfolded S k s d labeling (leftQuery S k g ω) =
        unfolded S k s d labeling (rightQuery S k g ω) := by
  rw [Constraint.satisfied, decide_eq_true_eq]
  have h := Encoding.translationTable_satisfied_iff
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))
    (vertexLabel S k s d labeling (vertex S k s d (leftQuery S k g ω)))
    (vertexLabel S k s d labeling (vertex S k s d (rightQuery S k g ω)))
  simp only [vertexLabel, Encoding.alphabetEquiv_apply_symm_apply] at h
  exact h.trans (ActualOrbit.constraint_iff_unfolded (canonical S k s d)
    (vertexLabel S k s d labeling) (vertexLabel S k s d labeling) _ _)

noncomputable def acceptanceProbability (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : ℚ :=
  𝔼 ω : Outcome S k g, if (edge S k g ω).satisfied labeling then (1 : ℚ) else 0

theorem acceptanceProbability_eq_test (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    acceptanceProbability S k g labeling =
      𝔼 ω : Outcome S k g,
        if unfolded S k s d labeling (leftQuery S k g ω) =
            unfolded S k s d labeling (rightQuery S k g ω)
        then (1 : ℚ) else 0 := by
  unfold acceptanceProbability
  apply Finset.expect_congr rfl
  intro ω _
  simp only [edge_satisfied_iff]

private theorem countSatisfied_map_inline_TableKeysGame {n q : Nat} {I : Type*}
    (labeling : Fin n → Fin q) (edges : I → Constraint n q) (xs : List I) :
    countSatisfied labeling (xs.map edges) =
      (xs.map (fun x => if (edges x).satisfied labeling then 1 else 0)).sum := by
  induction xs with
  | nil => rfl
  | cons x xs ih => simp only [List.map_cons, countSatisfied, List.sum_cons, ih]

theorem countSatisfied_eq_sum (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    countSatisfied labeling (outputInstance S k g).constraints =
      ∑ ω : Outcome S k g, if (edge S k g ω).satisfied labeling then 1 else 0 := by
  change countSatisfied labeling ((outcomes S k g).map (edge S k g)) = _
  rw [countSatisfied_map_inline_TableKeysGame]
  exact Finset.sum_map_toList _ _

theorem acceptanceProbability_eq_count (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    acceptanceProbability S k g labeling =
      (countSatisfied labeling (outputInstance S k g).constraints : ℚ) /
        (outputInstance S k g).constraints.length := by
  rw [acceptanceProbability, Fintype.expect_eq_sum_div_card,
    outputInstance_constraints_length, countSatisfied_eq_sum]
  congr 1
  simp

def explicitVertexCount (S : Source) (k s d : Nat) : Nat :=
  (ActualGame.explicitBodies S k s d).length

def explicitVertexEncoding (S : Source) (k s d : Nat) :
    Vertex S k s d ≃ Fin (explicitVertexCount S k s d) := by
  letI := ActualGame.orbitBodyDecidableEq S k s d
  exact ActualOrbit.explicitVertexEquiv (canonical S k s d)
    (ActualEnumeration.queries S.occurrences k s d) ActualEnumeration.mem_queries

noncomputable def semanticToExplicitVertices (S : Source) (k s d : Nat) :
    Fin (vertexCount S k s d) ≃ Fin (explicitVertexCount S k s d) :=
  (vertexEncoding S k s d).symm.trans (explicitVertexEncoding S k s d)

@[simp] theorem semanticToExplicitVertices_apply (S : Source) (k s d : Nat)
    (v : Vertex S k s d) :
    semanticToExplicitVertices S k s d (vertexEncoding S k s d v) =
      explicitVertexEncoding S k s d v := by
  simp [semanticToExplicitVertices]

theorem explicitVertexCount_eq (S : Source) (k s d : Nat) :
    explicitVertexCount S k s d = vertexCount S k s d := by
  simpa using (Fintype.card_congr (semanticToExplicitVertices S k s d)).symm

def explicitEdge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : Outcome S k g) : Constraint (explicitVertexCount S k s d) (2^s) where
  source := explicitVertexEncoding S k s d (vertex S k s d (leftQuery S k g ω))
  target := explicitVertexEncoding S k s d (vertex S k s d (rightQuery S k g ω))
  permutation := Encoding.translationTable
    (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω))

theorem explicitEdge_eq_rename (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : Outcome S k g) :
    explicitEdge S k g ω = Integration.InstanceEquivalences.renameConstraint
      (semanticToExplicitVertices S k s d) (edge S k g ω) := by
  simp only [explicitEdge, Integration.InstanceEquivalences.renameConstraint, edge,
    semanticToExplicitVertices_apply]

def outputInstanceWithEnumeration (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) : Instance (2^s) where
  vertices := explicitVertexCount S k s d
  constraints := (ActualGame.explicitOutcomes S k g en).map (explicitEdge S k g)
  nonempty := by
    intro h
    let ω : Outcome S k g := Classical.choice inferInstance
    have hm : explicitEdge S k g ω ∈
        (ActualGame.explicitOutcomes S k g en).map (explicitEdge S k g) :=
      List.mem_map.mpr ⟨ω, ActualGame.mem_explicitOutcomes S k g en ω, rfl⟩
    rw [h] at hm
    exact List.not_mem_nil hm

@[simp] theorem outputInstanceWithEnumeration_length (S : Source) (k : Nat)
    {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (outputInstanceWithEnumeration S k g en).constraints.length =
      Explicit.edgeCount S.occurrences k (s+d) en.indices.length := by
  simp only [outputInstanceWithEnumeration, List.length_map,
    ActualGame.explicitOutcomes, ActualEnumeration.length_indexedOutcomes]

theorem explicitConstraints_perm_renamed (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (outputInstanceWithEnumeration S k g en).constraints.Perm
      ((outputInstance S k g).constraints.map
        (Integration.InstanceEquivalences.renameConstraint (semanticToExplicitVertices S k s d))) := by
  simpa only [outputInstanceWithEnumeration, outputInstance, List.map_map,
    Function.comp_def, ← explicitEdge_eq_rename, outcomes] using
    (ActualGame.explicitOutcomes_perm_outcomes S k g en).map (explicitEdge S k g)

theorem completeAt_outputInstanceWithEnumeration_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    CompleteAt error (outputInstanceWithEnumeration S k g en) ↔
      CompleteAt error (outputInstance S k g) :=
  Integration.InstanceEquivalences.completeAt_iff_of_rename_perm error
    (outputInstance S k g) (outputInstanceWithEnumeration S k g en)
    (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en)

theorem soundAt_outputInstanceWithEnumeration_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    SoundAt error (outputInstanceWithEnumeration S k g en) ↔
      SoundAt error (outputInstance S k g) :=
  Integration.InstanceEquivalences.soundAt_iff_of_rename_perm error
    (outputInstance S k g) (outputInstanceWithEnumeration S k g en)
    (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en)

end DFVSGames.Decoder.TableKeysGame

namespace DFVSGames.Decoder.TableKeysRestoration

open DFVSGames.Integration.BinaryLinear
open DFVSGames.Reduction
open DFVSGames.Soundness
open ActualSource

noncomputable section
attribute [local instance] Classical.propDecidable

section OrbitExtension

variable {Q T : Type} {s d : Nat}

def extendedLabel (can : Q → Ambient s d × T)
    (label : ActualOrbit.Vertex can → Alphabet s) (body : Vector d × T) : Alphabet s :=
  if h : ∃ q, ActualOrbit.body can q = body then label ⟨body, h⟩ else 0

theorem extendedLabel_actual (can : Q → Ambient s d × T)
    (label : ActualOrbit.Vertex can → Alphabet s) (q : Q) :
    extendedLabel can label (ActualOrbit.body can q) = label (ActualOrbit.vertex can q) := by
  rw [extendedLabel, dite_eq_left ⟨q, rfl⟩]
  rfl

def restoredData (can : Q → Ambient s d × T)
    (label : ActualOrbit.Vertex can → Alphabet s) (D : Ambient s d × T) : Alphabet s :=
  extendedLabel can label (D.1.2, D.2) + D.1.1

theorem restoredData_actual (can : Q → Ambient s d × T)
    (label : ActualOrbit.Vertex can → Alphabet s) (q : Q) :
    restoredData can label (can q) = ActualOrbit.unfold can label q := by
  change extendedLabel can label (ActualOrbit.body can q) + ActualOrbit.offset can q = _
  rw [extendedLabel_actual]
  rfl

theorem restoredData_shift (can : Q → Ambient s d × T)
    (label : ActualOrbit.Vertex can → Alphabet s) (D : Ambient s d × T) (c : Alphabet s) :
    restoredData can label (D.1 + (c, 0), D.2) = restoredData can label D + c := by
  simp only [restoredData, Prod.fst_add, Prod.snd_add, add_zero]
  exact (add_assoc _ _ _).symm

end OrbitExtension

def keyAnswer (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (D : TableKeys.Key k (Fin S.«variables») (Fin S.occurrences) (Ambient s d)) : Alphabet s :=
  restoredData (TableKeysGame.canonical S k s d)
    (TableKeysGame.vertexLabel S k s d labeling) D

theorem keyAnswer_actual (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (q : TableKeysGame.Query S k s d) :
    keyAnswer S k s d labeling (TableKeysGame.canonical S k s d q) =
      TableKeysGame.unfolded S k s d labeling q :=
  restoredData_actual _ _ q

theorem keyAnswer_shift (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (D : TableKeys.Key k (Fin S.«variables») (Fin S.occurrences) (Ambient s d))
    (c : Alphabet s) :
    keyAnswer S k s d labeling (D.1 + (c, 0), D.2) = keyAnswer S k s d labeling D + c :=
  restoredData_shift _ _ D c

def projectedAnswer (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k))
    (O : RawPrivateTable.SupportedV J (ActualGame.names S))
    (Y : RawPartnerTarget.RawPoint J →ₗ[F2] Ambient s d) : Alphabet s :=
  keyAnswer S k s d labeling
    (TableKeys.projectedKey J (ActualGame.names S) (ActualGame.rhs S) O Y)

theorem projectedAnswer_pullback (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k)) (occ : Fin k → Fin S.occurrences)
    (slot : Fin k → PartnerProjection.Slot)
    (Y : RawPartnerTarget.RawPoint J →ₗ[F2] Ambient s d) :
    projectedAnswer S k s d labeling J
        (RawPrivateTable.supported J (ActualGame.names S) occ slot) Y =
      TableKeysGame.unfolded S k s d labeling
        (occ, Y.comp (RawPrivateTable.projection J (ActualGame.rhs S) occ slot)) := by
  unfold projectedAnswer
  rw [TableKeys.projectedKey_pullback]
  exact keyAnswer_actual S k s d labeling
    (occ, Y.comp (RawPrivateTable.projection J (ActualGame.rhs S) occ slot))

theorem projectedAnswer_fold (S : Source) (k s d : Nat)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k)) (O : RawPrivateTable.SupportedV J (ActualGame.names S))
    (Y : RawPartnerTarget.RawPoint J →ₗ[F2] Ambient s d) (c : Alphabet s) :
    projectedAnswer S k s d labeling J O
        (Y + (TableKeys.visibleTau J).smulRight (c, (0 : Vector d))) =
      projectedAnswer S k s d labeling J O Y + c := by
  unfold projectedAnswer
  rw [TableKeys.projectedKey_shift, keyAnswer_shift]

end

end DFVSGames.Decoder.TableKeysRestoration

namespace DFVSGames.Decoder.TableKeysPrivateRow

open DFVSGames.Integration.BinaryLinear
open DFVSGames.Reduction
open DFVSGames.Soundness
open ActualSource

noncomputable section

variable {R : Type} [AddCommGroup R] [Module F2 R]
variable {k s d : Nat}

theorem assemble_prod_shift (J : Finset (Fin k)) (A : Alphabet s →ₗ[F2] R)
    (M₀ : RawPartnerTarget.RawPoint J →ₗ[F2] Alphabet s)
    (N : RawPartnerTarget.RawPoint J →ₗ[F2] A.ker)
    (T : RawPartnerTarget.RawPoint J →ₗ[F2] Vector d) (h : A.ker) :
    (AdviceFibers.assemble A M₀ (N + (TableKeys.visibleTau J).smulRight h)).prod T =
      (AdviceFibers.assemble A M₀ N).prod T +
        (TableKeys.visibleTau J).smulRight ((h : Alphabet s), (0 : Vector d)) := by
  apply LinearMap.ext
  intro x
  apply Prod.ext
  · change M₀ x + ((N x : Alphabet s) + TableKeys.visibleTau J x • (h : Alphabet s)) =
      (M₀ x + (N x : Alphabet s)) + TableKeys.visibleTau J x • (h : Alphabet s)
    exact (add_assoc _ _ _).symm
  · change T x = T x + TableKeys.visibleTau J x • (0 : Vector d)
    simp

def rowAnswer (S : Source)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k))
    (Q : RawPrivateTable.SupportedV J (ActualGame.names S))
    (A : Alphabet s →ₗ[F2] R)
    (rows : AdviceFibers.ObservedRow (E := RawPartnerTarget.RawPoint J) A)
    (T : RawPartnerTarget.RawPoint J →ₗ[F2] Vector d)
    (N : RawPartnerTarget.RawPoint J →ₗ[F2] A.ker) : Alphabet s :=
  TableKeysRestoration.projectedAnswer S k s d labeling J Q
    ((AdviceFibers.assemble A (AdviceFibers.observedBase A rows) N).prod T)

theorem rowAnswer_fold (S : Source)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k)) (Q : RawPrivateTable.SupportedV J (ActualGame.names S))
    (A : Alphabet s →ₗ[F2] R)
    (rows : AdviceFibers.ObservedRow (E := RawPartnerTarget.RawPoint J) A)
    (T : RawPartnerTarget.RawPoint J →ₗ[F2] Vector d)
    (N : RawPartnerTarget.RawPoint J →ₗ[F2] A.ker) (h : A.ker) :
    rowAnswer S labeling J Q A rows T (N + (TableKeys.visibleTau J).smulRight h) =
      rowAnswer S labeling J Q A rows T N + (h : Alphabet s) := by
  unfold rowAnswer
  rw [assemble_prod_shift, TableKeysRestoration.projectedAnswer_fold]

theorem rowAnswer_actual (S : Source)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2^s))
    (J : Finset (Fin k)) (Q : RawPrivateTable.SupportedV J (ActualGame.names S))
    (A : Alphabet s →ₗ[F2] R)
    (M : RawPartnerTarget.RawPoint J →ₗ[F2] Alphabet s)
    (T : RawPartnerTarget.RawPoint J →ₗ[F2] Vector d) :
    rowAnswer S labeling J Q A (AdviceFibers.observe A M) T
        (AdviceFibers.hiddenCoordinate A M) =
      TableKeysRestoration.projectedAnswer S k s d labeling J Q (M.prod T) := by
  have h := (AdviceFibers.observationEquiv A).left_inv M
  change AdviceFibers.assemble A (AdviceFibers.observedBase A (AdviceFibers.observe A M))
      (AdviceFibers.hiddenCoordinate A M) = M at h
  unfold rowAnswer
  rw [h]

def targetIntercept (J : Finset (Fin k)) (A : Alphabet s →ₗ[F2] R)
    (rows : AdviceFibers.ObservedRow (E := RawPartnerTarget.RawPoint J) A)
    (z : RawPartnerTarget.RawPoint J) (u : Alphabet s) : Alphabet s :=
  AdviceFibers.observedBase A rows z + u

theorem affine_target_on_row (J : Finset (Fin k)) (A : Alphabet s →ₗ[F2] R)
    (rows : AdviceFibers.ObservedRow (E := RawPartnerTarget.RawPoint J) A)
    (N : RawPartnerTarget.RawPoint J →ₗ[F2] A.ker)
    (z : RawPartnerTarget.RawPoint J) (u : Alphabet s) :
    AdviceFibers.assemble A (AdviceFibers.observedBase A rows) N z + u =
      (N z : Alphabet s) + targetIntercept J A rows z u :=
  AdviceFibers.assemble_affine_target A (AdviceFibers.observedBase A rows) N z u

end

end DFVSGames.Decoder.TableKeysPrivateRow

end OAI