-- Clean/Circuit/Theorems.lean

import Clean.Circuit.Basic

import Clean.Circuit.Provable

variable {F : Type} [Field F] {α β : Type}

namespace Operations

end Operations

namespace Circuit

end Circuit

namespace FlatOperation

end FlatOperation

namespace Environment

open FlatOperation (localLength localWitnesses)

end Environment

namespace Circuit

end Circuit

namespace Circuit

variable {α β : Type} {n : ℕ} {prop : Condition F} {env : Environment F}

end Circuit

namespace FlatOperation

end FlatOperation

namespace Operations

-- Repository definitions used:
content: class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where
  main : Var Input F → Circuit F (Var Output F)

   
  localLength : Var Input F → ℕ

   
  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input
    := by admit /- proof elided -/

 @[circuit_norm]
def forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop
  | [] => True
  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops
  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops
  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops
  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops

content: structure Condition (F : Type) [Field F] where
  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True
  assert (offset : ℕ) (_ : Expression F) : Prop := True
  lookup (offset : ℕ) (_ : Lookup F) : Prop := True
  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True

content: inductive FlatOperation (F : Type) where
  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F
  | assert : Expression F → FlatOperation F
  | lookup : Lookup F → FlatOperation F

content: @[circuit_norm]
def localLength : Operation F → ℕ
  | .witness m _ => m
  | .assert _ => 0
  | .lookup _ => 0
  | .subcircuit s => s.localLength

content: @[circuit_norm]
def localLength : List (FlatOperation F) → ℕ
  | [] => 0
  | witness m _ :: ops => m + localLength ops
  | assert _ :: ops | lookup _ :: ops => localLength ops

content: @[circuit_norm]
def lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>
  ((), [.lookup { table := table.toRaw, entry := toElements entry }])

content: def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)

content: inductive Operation (F : Type) [Field F] where
  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F
  | assert : Expression F → Operation F
  | lookup : Lookup F → Operation F
  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F

content: structure Subcircuit (F : Type) [Field F] (offset : ℕ) where
  ops : List (FlatOperation F)

  
  
  
  Soundness : Environment F → Prop
  Completeness : Environment F → Prop
  UsesLocalWitnesses : Environment F → Prop

  
  
  localLength : ℕ

  
  imply_soundness : ∀ env,
    ConstraintsHoldFlat env ops → Soundness env

  
  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →
    Completeness env → ConstraintsHoldFlat env ops

  
  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →
    UsesLocalWitnesses env

  
  localLength_eq : localLength = FlatOperation.localLength ops

content: @[circuit_norm]
def localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)
  | [] => #v[]
  | witness _ compute :: ops => compute env ++ localWitnesses env ops
  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops

content: structure Environment (F : Type) where
  get : ℕ → F

content: def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop
  | [] => True
  | op :: ops => match op with
    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops
    | lookup { table, entry } =>
      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops
    | _ => ConstraintsHoldFlat eval ops

content: structure Lookup (F : Type) where
  table : RawTable F
  entry : Vector (Expression F) table.arity

content: structure RawTable (F : Type) where
  name : String
  arity : ℕ
  Contains : Vector F arity → Prop
  Soundness : Vector F arity → Prop
  Completeness : Vector F arity → Prop
  imply_soundness : ∀ row, Contains row → Soundness row
  implied_by_completeness : ∀ row, Completeness row → Contains row

content: structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where
  name : String
  length : ℕ
  row : Fin length → Row F
  
  
  
  index : Row F → ℕ
  Spec : Row F → Prop
  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t

content: def Contains (table : StaticTable F Row) (row : Row F) :=
  ∃ i : Fin table.length, row = table.row i

content: instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where
  size := combinedSize α
  toElements x :=
    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm
  fromElements v :=
    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents
  fromElements_toElements x := by admit /- proof elided -/

content: @[reducible]
def TypeMap := Type → Type

content: inductive Expression (F : Type) where
  | var : Variable F -> Expression F
  | const : F -> Expression F
  | add : Expression F -> Expression F -> Expression F
  | mul : Expression F -> Expression F -> Expression F

content: structure Variable (F : Type) where
  index : ℕ

content: def const (x : α F) : Var α F :=
  let values : Vector F _ := toElements x
  fromVars (values.map .const)

content: @[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)

content: @[circuit_norm]
def fromVars (vars : Vector (Expression F) (size M)) := fromElements vars

content: class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where
  witness : ((Environment F) → value F) → Circuit F (var F)
  var_eq : var F = value (Expression F) := by admit /- proof elided -/

content: @[circuit_norm]
def Table.toRaw (table : Table F Row) : RawTable F where
  name := table.name
  arity := size Row
  Contains row := table.Contains (fromElements row)
  Soundness row := table.Soundness (fromElements row)
  Completeness row := table.Completeness (fromElements row)
  imply_soundness row := table.imply_soundness (fromElements row)
  implied_by_completeness row := table.implied_by_completeness (fromElements row)

content: structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where
  name : String
   
  Contains : Row F → Prop

   
  Soundness : Row F → Prop
  Completeness : Row F → Prop

  imply_soundness : ∀ row, Contains row → Soundness row
  implied_by_completeness : ∀ row, Completeness row → Contains row

content: @[circuit_norm]
def localLength : Operations F → ℕ
  | [] => 0
  | .witness m _ :: ops => m + localLength ops
  | .assert _ :: ops => localLength ops
  | .lookup _ :: ops => localLength ops
  | .subcircuit s :: ops => s.localLength + localLength ops

content: @[reducible, circuit_norm]
def Operations (F : Type) [Field F] := List (Operation F)

content: def toFlat : Operations F → List (FlatOperation F)
  | [] => []
  | .witness m c :: ops => .witness m c :: toFlat ops
  | .assert e :: ops => .assert e :: toFlat ops
  | .lookup l :: ops => .lookup l :: toFlat ops
  | .subcircuit s :: ops => s.ops ++ toFlat ops

content: def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=
  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops

content: def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop
  | [] => True
  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops
  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops
  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops

content: class ConstantLength (circuit : α → Circuit F β) where
  localLength : ℕ
  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength

content: def induct {motive : Operations F → Sort*}
  (empty : motive [])
  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))
  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))
  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))
  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))
    (ops : Operations F) : motive ops :=
  match ops with
  | [] => empty
  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)
  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)
  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)
  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)

content: def FlatOperation.singleLocalLength : FlatOperation F → ℕ
  | .witness m _ => m
  | .assert _ => 0
  | .lookup _ => 0

content: def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop
  | .witness m c => condition.witness offset m c
  | .assert e => condition.assert offset e
  | .lookup l => condition.lookup offset l

-- Repository lemmas used:
content: theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]

-- Local definitions used:

-- Local lemmas used:
content: lemma localLength_cons {F} {op : FlatOperation F} {ops : List (FlatOperation F)} :
    localLength (op :: ops) = op.singleLocalLength + localLength ops

content: theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :
  forAll offset condition (op :: ops) ↔
    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops

content: lemma forAll_append {condition : Condition F} {ops ops' : List (FlatOperation F)} (n : ℕ) :
  forAll n condition (ops ++ ops') ↔
    forAll n condition ops ∧ forAll (localLength ops + n) condition ops'

-- Target theorem and proof:
lemma forAll_toFlat_iff (n : ℕ) (condition : Condition F) (ops : Operations F) :
    FlatOperation.forAll n condition ops.toFlat ↔ ops.forAllFlat n condition
:= by
  induction ops using Operations.induct generalizing n with
  | empty => simp only [forAllFlat, forAll, toFlat, FlatOperation.forAll]
  | witness | assert | lookup =>
    simp_all [forAllFlat, forAll, toFlat, FlatOperation.forAll]
  | subcircuit s ops ih =>
    simp_all only [forAllFlat, forAll, toFlat]
    rw [FlatOperation.forAll_append, s.localLength_eq]
    simp_all