/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

NGemetry.lean — N-Dimensional Geometry Extension

Extends SpatialEvo from 3D to n-dimensional geometry for VLSI design
and general spatial reasoning applications.

Key contributions:
1. Generic PointND structure for n-dimensional points
2. Generic VectorND structure for n-dimensional vectors
3. N-dimensional spatial algorithms (distance, ordering, orientation)
4. N-dimensional camera pose and scene representation
5. Verification examples and theorems

Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: All defs must have eval witnesses or theorems.
-/

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Array.Basic

namespace Semantics.NGemetry

-- ════════════════════════════════════════════════════════════
-- §0  Fixed-Point Precision (Q16.16 for n-dimensional computations)
-- ════════════════════════════════════════════════════════════

/-- Q16.16 fixed-point for n-dimensional geometry. -/
structure Q1616 where
  raw : Int
  deriving Repr, DecidableEq, Inhabited, BEq

namespace Q1616

def zero    : Q1616 := ⟨0⟩
def one     : Q1616 := ⟨65536⟩        -- 0x00010000 = 1.0

def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩

def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
def sub (a b : Q1616) : Q1616 := ⟨a.raw - b.raw⟩
def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
def div (a b : Q1616) : Q1616 := ⟨(a.raw * 65536) / b.raw⟩

instance : Add Q1616 := ⟨add⟩
instance : Sub Q1616 := ⟨sub⟩
instance : Mul Q1616 := ⟨mul⟩
instance : Div Q1616 := ⟨div⟩
instance : Neg Q1616 := ⟨fun a => ⟨-a.raw⟩⟩

instance : LE Q1616 := ⟨fun a b => a.raw ≤ b.raw⟩
instance : LT Q1616 := ⟨fun a b => a.raw < b.raw⟩

/-- Absolute value. -/
def abs (a : Q1616) : Q1616 := if a.raw < 0 then ⟨-a.raw⟩ else a

/-- Minimum of two values. -/
def min (a b : Q1616) : Q1616 := if a ≤ b then a else b

/-- Maximum of two values. -/
def max (a b : Q1616) : Q1616 := if a ≥ b then a else b

end Q1616

-- ════════════════════════════════════════════════════════════
-- §1  N-Dimensional Point and Vector Structures
-- ════════════════════════════════════════════════════════════

/-- N-dimensional point in space. -/
structure PointND (n : Nat) where
  coordinates : Array Q1616
  dimension : Nat := n
  hDim : dimension = n
  deriving Repr, Inhabited

namespace PointND

/-- Create point from array of coordinates. -/
def fromArray (coords : Array Q1616) (n : Nat) : PointND n :=
  { coordinates := coords, dimension := n, hDim := by simp }

/-- Get coordinate at index i. -/
def getCoord (p : PointND n) (i : Nat) (h : i < n) : Q1616 :=
  p.coordinates.get ⟨i, h⟩

/-- Euclidean distance between two n-dimensional points. -/
def euclideanDistance (p1 p2 : PointND n) : Q1616 :=
  let n := p1.dimension
  let sumSquared := (List.range n).foldl (fun acc i =>
    let c1 := p1.getCoord i (by simp_arith [h₁])
    let c2 := p2.getCoord i (by simp_arith [h₂])
    let diff := Q1616.sub c1 c2
    let squared := Q1616.mul diff diff
    Q1616.add acc squared
  ) Q1616.zero
  -- Compute square root (simplified as identity for Q16.16)
  sumSquared

/-- Manhattan distance between two n-dimensional points. -/
def manhattanDistance (p1 p2 : PointND n) : Q1616 :=
  let n := p1.dimension
  (List.range n).foldl (fun acc i =>
    let c1 := p1.getCoord i (by simp_arith [h₁])
    let c2 := p2.getCoord i (by simp_arith [h₂])
    let diff := Q1616.sub c1 c2
    let absDiff := Q1616.abs diff
    Q1616.add acc absDiff
  ) Q1616.zero

/-- Origin point in n-dimensional space. -/
def origin (n : Nat) : PointND n :=
  fromArray (Array.mkArray n Q1616.zero) n

end PointND

/-- N-dimensional vector in space. -/
structure VectorND (n : Nat) where
  components : Array Q1616
  dimension : Nat := n
  hDim : dimension = n
  deriving Repr, Inhabited

namespace VectorND

/-- Create vector from array of components. -/
def fromArray (comps : Array Q1616) (n : Nat) : VectorND n :=
  { components := comps, dimension := n, hDim := by simp }

/-- Get component at index i. -/
def getComp (v : VectorND n) (i : Nat) (h : i < n) : Q1616 :=
  v.components.get ⟨i, h⟩

/-- Vector addition. -/
def add (v1 v2 : VectorND n) : VectorND n :=
  let n := v1.dimension
  let newComps := (List.range n).map (fun i =>
    let c1 := v1.getComp i (by simp_arith [h₁])
    let c2 := v2.getComp i (by simp_arith [h₂])
    Q1616.add c1 c2
  )
  fromArray newComps n

/-- Vector subtraction. -/
def sub (v1 v2 : VectorND n) : VectorND n :=
  let n := v1.dimension
  let newComps := (List.range n).map (fun i =>
    let c1 := v1.getComp i (by simp_arith [h₁])
    let c2 := v2.getComp i (by simp_arith [h₂])
    Q1616.sub c1 c2
  )
  fromArray newComps n

/-- Dot product of two n-dimensional vectors. -/
def dot (v1 v2 : VectorND n) : Q1616 :=
  let n := v1.dimension
  (List.range n).foldl (fun acc i =>
    let c1 := v1.getComp i (by simp_arith [h₁])
    let c2 := v2.getComp i (by simp_arith [h₂])
    let prod := Q1616.mul c1 c2
    Q1616.add acc prod
  ) Q1616.zero

/-- Vector magnitude (Euclidean norm). -/
def magnitude (v : VectorND n) : Q1616 :=
  let dotProd := dot v v
  -- Square root (simplified as identity for Q16.16)
  dotProd

/-- Normalize vector to unit length. -/
def normalize (v : VectorND n) : VectorND n :=
  let mag := magnitude v
  let n := v.dimension
  if mag = Q1616.zero then
    v  -- Return zero vector unchanged
  else
    let newComps := (List.range n).map (fun i =>
      let c := v.getComp i (by simp_arith [h])
      Q1616.div c mag
    )
    fromArray newComps n

/-- Zero vector in n-dimensional space. -/
def zero (n : Nat) : VectorND n :=
  fromArray (Array.mkArray n Q1616.zero) n

end VectorND

-- ════════════════════════════════════════════════════════════
-- §2  N-Dimensional Camera and Scene Structures
-- ════════════════════════════════════════════════════════════

/-- N-dimensional camera pose (position + orientation). -/
structure CameraPoseND (n : Nat) where
  position : PointND n
  rotation : VectorND n  -- Simplified: n-dimensional rotation parameters
  frameIndex : Nat
  deriving Repr, Inhabited

/-- N-dimensional point cloud with density metric. -/
structure PointCloudND (n : Nat) where
  points : Array (PointND n)
  density : Q1616  -- Points per unit volume
  dimension : Nat := n
  deriving Repr, Inhabited

/-- N-dimensional bounding hyperbox. -/
struct BoundingHyperbox (n : Nat) where
  min : PointND n
  max : PointND n
  deriving Repr, Inhabited

/-- N-dimensional scene containing geometric assets. -/
structure SceneND (n : Nat) where
  name : String
  pointCloud : PointCloudND n
  cameraPoses : Array (CameraPoseND n)
  objects : Array (BoundingHyperbox n)
  deriving Repr, Inhabited

-- ════════════════════════════════════════════════════════════
-- §3  N-Dimensional Spatial Algorithms
-- ════════════════════════════════════════════════════════════

/-- Compute camera orientation between two n-dimensional poses. -/
def computeCameraOrientationND (n : Nat) (pose1 pose2 : CameraPoseND n) : VectorND n :=
  VectorND.sub pose2.position pose1.position

/-- Compute depth ordering for n-dimensional objects. -/
def computeDepthOrderingND (n : Nat) (camera : PointND n) (objects : Array (BoundingHyperbox n)) : Array Nat :=
  let distances := objects.mapIdx (fun i obj =>
    let center := PointND.fromArray 
      (Array.mkArray n (Q1616.div (Q1616.add obj.min.getCoord 0 (by sorry) obj.max.getCoord 0 (by sorry)) Q1616.one)) n
    let dist := PointND.euclideanDistance camera center
    (i, dist)
  )
  distances.toArray.map (fun p => p.1)

/-- Compute object distance in n-dimensional space. -/
def computeObjectDistanceND (n : Nat) (obj1 obj2 : BoundingHyperbox n) : Q1616 :=
  let center1 := PointND.fromArray 
    (Array.mkArray n (Q1616.div (Q1616.add obj1.min.getCoord 0 (by sorry) obj1.max.getCoord 0 (by sorry)) Q1616.one)) n
  let center2 := PointND.fromArray 
    (Array.mkArray n (Q1616.div (Q1616.add obj2.min.getCoord 0 (by sorry) obj2.max.getCoord 0 (by sorry)) Q1616.one)) n
  PointND.euclideanDistance center1 center2

/-- Check if two n-dimensional bounding hyperboxes intersect. -/
def hyperboxIntersection (n : Nat) (box1 box2 : BoundingHyperbox n) : Bool :=
  -- Simplified: check if any dimension overlaps
  false  -- TODO(lean-port): Implement proper n-dimensional intersection test

-- ════════════════════════════════════════════════════════════
-- §4  Theorems: N-Dimensional Geometry Properties
-- ════════════════════════════════════════════════════════════

/-- Theorem: Origin point has zero distance to itself. -/
theorem originDistanceZero (n : Nat) :
    PointND.euclideanDistance (PointND.origin n) (PointND.origin n) = Q1616.zero := by
  sorry  -- TODO(lean-port): Prove origin distance is zero

/-- Theorem: Euclidean distance is symmetric. -/
theorem euclideanDistanceSymmetric (n : Nat) (p1 p2 : PointND n) :
    PointND.euclideanDistance p1 p2 = PointND.euclideanDistance p2 p1 := by
  sorry  -- TODO(lean-port): Prove Euclidean distance symmetry

/-- Theorem: Manhattan distance satisfies triangle inequality. -/
theorem manhattanTriangleInequality (n : Nat) (p1 p2 p3 : PointND n) :
    let d12 := PointND.manhattanDistance p1 p2
    let d23 := PointND.manhattanDistance p2 p3
    let d13 := PointND.manhattanDistance p1 p3
    d13 ≤ d12 + d23 := by
  sorry  -- TODO(lean-port): Prove Manhattan triangle inequality

/-- Theorem: Dot product is commutative. -/
theorem dotProductCommutative (n : Nat) (v1 v2 : VectorND n) :
    VectorND.dot v1 v2 = VectorND.dot v2 v1 := by
  sorry  -- TODO(lean-port): Prove dot product commutativity

/-- Theorem: Zero vector has zero magnitude. -/
theorem zeroVectorMagnitude (n : Nat) :
    VectorND.magnitude (VectorND.zero n) = Q1616.zero := by
  sorry  -- TODO(lean-port): Prove zero vector has zero magnitude

-- ════════════════════════════════════════════════════════════
-- §5  Verification Examples
-- ════════════════════════════════════════════════════════════

#eval PointND.origin 3  -- Expected: Point with 3 zero coordinates

#eval let p1 := PointND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 2, Q1616.ofNat 3]) 3
      let p2 := PointND.fromArray (#[Q1616.ofNat 4, Q1616.ofNat 5, Q1616.ofNat 6]) 3
      PointND.euclideanDistance p1 p2  -- Expected: distance between points

#eval let v := VectorND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 0, Q1616.ofNat 0]) 3
      VectorND.magnitude v  -- Expected: magnitude of vector

#eval let v1 := VectorND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 2, Q1616.ofNat 3]) 3
      let v2 := VectorND.fromArray (#[Q1616.ofNat 4, Q1616.ofNat 5, Q1616.ofNat 6]) 3
      VectorND.dot v1 v2  -- Expected: dot product

-- TODO(lean-port): Add n-dimensional camera orientation example
-- TODO(lean-port): Add n-dimensional depth ordering example

end Semantics.NGemetry