import Mathlib

namespace OAI

open scoped BigOperators ENNReal NNReal Topology ContDiff
open MeasureTheory
namespace AlternatingNS

theorem neZeroThree : NeZero 3 := ⟨Nat.succ_ne_zero 2⟩

theorem twoAtLeastTwo : Nat.AtLeastTwo 2 := ⟨Nat.le_refl 2⟩

theorem threeAtLeastTwo : Nat.AtLeastTwo 3 := ⟨Nat.le_succ 2⟩

theorem eightAtLeastTwo : Nat.AtLeastTwo 8 := ⟨by decide⟩

abbrev Rule := ℕ × ℕ × Fin 3

abbrev Machine := ℕ × ℕ × ℕ × List ℕ × List (Option Rule)

namespace Machine

def states (M : Machine) : ℕ := M.1 + 1

def alphabet (M : Machine) : ℕ := M.2.1 + 1

def initialState (M : Machine) : ℕ := M.2.2.1

def haltingStates (M : Machine) : List ℕ := M.2.2.2.1

def table (M : Machine) : List (Option Rule) := M.2.2.2.2

def instruction (M : Machine) (q a : ℕ) : Option Rule :=
  (M.table[q * M.alphabet + a]?).join

def WellFormed (M : Machine) : Prop :=
  M.initialState < M.states ∧
  (∀ q ∈ M.haltingStates, q < M.states) ∧
  ∀ q a r, q < M.states → a < M.alphabet → M.instruction q a = some r →
    r.1 < M.states ∧ r.2.1 < M.alphabet

def ValidInput (M : Machine) (w : List ℕ) : Prop :=
  ∀ a ∈ w, a < M.alphabet

structure Configuration where
  state : ℕ
  head : ℤ
  tape : ℤ → ℕ

def isHalting (M : Machine) (c : Configuration) : Bool :=
  decide (c.state = M.states ∨ c.state ∈ M.haltingStates)

def initial (M : Machine) (w : List ℕ) : Configuration where
  state := M.initialState
  head := 0
  tape := fun i => if 0 ≤ i then w[i.toNat]?.getD 0 else 0

def step (M : Machine) (c : Configuration) : Configuration :=
  if M.isHalting c then c else
    match M.instruction c.state (c.tape c.head) with
    | none => { c with state := M.states }
    | some r =>
      { state := r.1
        head := c.head + ((r.2.2.val : ℤ) - 1)
        tape := Function.update c.tape c.head r.2.1 }

def run (M : Machine) (w : List ℕ) (n : ℕ) : Configuration :=
  (M.step)^[n] (M.initial w)

def Halts (M : Machine) (w : List ℕ) : Prop :=
  ∃ n : ℕ, M.isHalting (M.run w n) = true

end Machine

abbrev Space := EuclideanSpace ℝ (Fin 3)
abbrev Field (E : Type*) := ℝ → Space → E
abbrev Velocity := Field Space
abbrev Pressure := Field ℝ
abbrev MultiIndex := Fin 3 → ℕ

noncomputable section

def e (i : Fin 3) : Space := EuclideanSpace.single i 1

def dx {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (i : Fin 3) (F : Field E) : Field E :=
  fun t x => fderiv ℝ (F t) x (e i)

def dt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (F : Field E) : Field E :=
  fun t x => fderivWithin ℝ (fun s => F s x) (Set.Ici 0) t 1

def spatial {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (α : MultiIndex) (F : Field E) : Field E :=
  (dx 0)^[α 0] ((dx 1)^[α 1] ((dx 2)^[α 2] F))

def mixed {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (l : ℕ) (α : MultiIndex) (F : Field E) : Field E :=
  dt^[l] (spatial α F)

def div (u : Velocity) (t : ℝ) (x : Space) : ℝ :=
  ∑ i : Fin 3, dx i u t x i

def grad (p : Pressure) (t : ℝ) (x : Space) : Space :=
  (EuclideanSpace.equiv (Fin 3) ℝ).symm (fun i => dx i p t x)

def laplacian (u : Velocity) (t : ℝ) (x : Space) : Space :=
  ∑ i : Fin 3, dx i (dx i u) t x

def advection (u : Velocity) (t : ℝ) (x : Space) : Space :=
  fderiv ℝ (u t) x (u t x)

def residual (ν : ℝ) (u : Velocity) : Velocity :=
  fun t x => dt u t x + advection u t x - ν • laplacian u t x

def timeCylinder (T : ℝ) : Set (ℝ × Space) := Set.Icc 0 T ×ˢ Set.univ

def Smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (F : Field E) : Prop :=
  ContDiffOn ℝ (∞) (Function.uncurry F) (Set.Ici 0 ×ˢ Set.univ)

def ClassicalRegularity (u : Velocity) (p : Pressure) : Prop :=
  (∀ x, DifferentiableOn ℝ (fun t => u t x) (Set.Ici 0)) ∧
  (∀ t, 0 ≤ t → Differentiable ℝ (u t)) ∧
  (∀ t, 0 ≤ t → ∀ i, Differentiable ℝ (dx i u t)) ∧
  (∀ t, 0 ≤ t → Differentiable ℝ (p t)) ∧
  ∀ T, 0 ≤ T →
    ContinuousOn (Function.uncurry u) (timeCylinder T) ∧
    ContinuousOn (Function.uncurry (dt u)) (timeCylinder T) ∧
    (∀ i, ContinuousOn (Function.uncurry (dx i u)) (timeCylinder T)) ∧
    (∀ i j, ContinuousOn (Function.uncurry (dx i (dx j u))) (timeCylinder T)) ∧
    ContinuousOn (Function.uncurry p) (timeCylinder T) ∧
    (∀ i, ContinuousOn (Function.uncurry (dx i p)) (timeCylinder T))

def ContinuousInH {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (k : ℕ) (F : Field E) (T : ℝ) : Prop :=
  ∀ α : MultiIndex, (∑ i, α i) ≤ k →
    ∃ g : ℝ → Lp E 2 (volume : Measure Space),
      ContinuousOn g (Set.Icc 0 T) ∧
      ∀ t ∈ Set.Icc 0 T,
        ∀ᵐ x ∂(volume : Measure Space), (g t) x = spatial α F t x

def C1InL2 (u : Velocity) (T : ℝ) : Prop :=
  ∃ g g' : ℝ → Lp Space 2 (volume : Measure Space),
    ContinuousOn g (Set.Icc 0 T) ∧
    ContinuousOn g' (Set.Icc 0 T) ∧
    (∀ t ∈ Set.Icc 0 T, HasDerivWithinAt g (g' t) (Set.Icc 0 T) t) ∧
    (∀ t ∈ Set.Icc 0 T, ∀ᵐ x ∂(volume : Measure Space), (g t) x = u t x) ∧
    (∀ t ∈ Set.Icc 0 T, ∀ᵐ x ∂(volume : Measure Space), (g' t) x = dt u t x)

def CompetitorEnergyClass (u : Velocity) (p : Pressure) : Prop :=
  ∀ T, 0 ≤ T →
    ContinuousInH 2 u T ∧ C1InL2 u T ∧ ContinuousInH 1 p T ∧
    ∃ C : ℝ, ∀ t ∈ Set.Icc 0 T, ∀ x, ‖u t x‖ ≤ C

def EnergyClass (u : Velocity) (p : Pressure) : Prop :=
  CompetitorEnergyClass u p ∧
  ∀ T, 0 ≤ T → ∃ C : ℝ, ∀ t ∈ Set.Icc 0 T, ∀ x,
    ‖u t x‖ + ‖fderiv ℝ (u t) x‖ ≤ C

def NavierStokes (ν : ℝ) (f u : Velocity) (p : Pressure) : Prop :=
  ClassicalRegularity u p ∧
  (∀ x, u 0 x = 0) ∧
  ∀ t, 0 ≤ t → ∀ x,
    div u t x = 0 ∧
    dt u t x + advection u t x = -grad p t x + ν • laplacian u t x + f t x

def Rapid (F : Velocity) : Prop :=
  ∀ J l : ℕ, ∀ α : MultiIndex, ∃ C : ℝ, 0 ≤ C ∧
    ∀ t, 0 ≤ t → ∀ x,
      (1 + t + ‖x‖) ^ J * ‖mixed l α F t x‖ ≤ C

def SupportedIn (F : Velocity) (K : Set Space) : Prop :=
  ∀ t, 0 ≤ t → Function.support (F t) ⊆ K

def IsTrajectory (u : Velocity) (a : Space) (γ : ℝ → Space) : Prop :=
  γ 0 = a ∧ ∀ t, 0 ≤ t → HasDerivWithinAt γ (u t (γ t)) (Set.Ici 0) t

def observedParticle : Space := (EuclideanSpace.equiv (Fin 3) ℝ).symm ![-1, 0, 0]

abbrev RationalPoint := ℚ × ℚ × ℚ × ℚ
abbrev RationalVector := ℚ × ℚ × ℚ
abbrev IndexCode := ℕ × ℕ × ℕ
abbrev Program := Nat.Partrec.Code
abbrev FieldProgram := Program × Program × Program

def Returns {A B : Type*} [Encodable A] [Encodable B]
    (c : Program) (a : A) (b : B) : Prop :=
  Encodable.encode b ∈ c.eval (Encodable.encode a)

def index (α : IndexCode) : MultiIndex := ![α.1, α.2.1, α.2.2]

def rationalVector (q : RationalVector) : Space :=
  (EuclideanSpace.equiv (Fin 3) ℝ).symm ![(q.1 : ℝ), (q.2.1 : ℝ), (q.2.2 : ℝ)]

def rationalPoint (q : RationalPoint) : ℝ × Space :=
  ((q.1 : ℝ), rationalVector q.2)

def dyadicError (n : ℕ) : ℝ := (1 / 2 : ℝ) ^ n

def ComputableReal (r : ℝ) : Prop :=
  ∃ a : ℕ → ℚ, Computable (fun n => ((a n).num, (a n).den)) ∧
    ∀ n, |r - (a n : ℝ)| ≤ dyadicError n

def ProgramFor (c : FieldProgram) (F : Velocity) : Prop :=
  (∀ (l : ℕ) (α : IndexCode) (q : RationalPoint) (k : ℕ),
    ∃ v : RationalVector, Returns c.1 (l, α, q, k) v ∧
      (0 ≤ q.1 →
        ‖rationalVector v - mixed l (index α) F (rationalPoint q).1 (rationalPoint q).2‖
          ≤ dyadicError k)) ∧
  (∀ (l : ℕ) (α : IndexCode) (R k : ℕ),
    ∃ m : ℕ, Returns c.2.1 (l, α, R, k) m ∧
      ∀ z z' : ℝ × Space,
        0 ≤ z.1 → 0 ≤ z'.1 → ‖z‖ ≤ R → ‖z'‖ ≤ R →
        ‖z - z'‖ ≤ dyadicError m →
        ‖mixed l (index α) F z.1 z.2 - mixed l (index α) F z'.1 z'.2‖ ≤ dyadicError k) ∧
  (∀ (J l : ℕ) (α : IndexCode),
    ∃ B : ℕ, Returns c.2.2 (J, l, α) B ∧
      ∀ t, 0 ≤ t → ∀ x,
        (1 + t + ‖x‖) ^ J * ‖mixed l (index α) F t x‖ ≤ B)

def AlternatingConclusion (ν : ℝ) : Prop :=
  ∃ (compileForce compileVelocity : Machine × List ℕ → FieldProgram),
    Computable compileForce ∧ Computable compileVelocity ∧
    ∃ K : Set Space, IsCompact K ∧
      ∀ (M : Machine) (w : List ℕ), M.WellFormed → M.ValidInput w →
        ∃ (f U : Velocity) (X : Space → ℝ → Space),
          ProgramFor (compileForce (M, w)) f ∧
          ProgramFor (compileVelocity (M, w)) U ∧
          Smooth f ∧ Smooth U ∧
          SupportedIn f K ∧ SupportedIn U K ∧ Rapid f ∧ Rapid U ∧
          f = residual ν U ∧
          NavierStokes ν f U (fun _ _ => 0) ∧ EnergyClass U (fun _ _ => 0) ∧
          (∀ (v : Velocity) (p : Pressure),
            NavierStokes ν f v p → CompetitorEnergyClass v p →
            ∀ t, 0 ≤ t → ∀ x, v t x = U t x ∧ p t x = 0) ∧
          (∀ a, IsTrajectory U a (X a)) ∧
          (∀ a γ, IsTrajectory U a γ → ∀ t, 0 ≤ t → γ t = X a t) ∧
          ((∃ t : ℝ, 0 ≤ t ∧ 0 < X observedParticle t 0) ↔ M.Halts w)

end
end AlternatingNS

end OAI