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Replace terse -- TODO(lean-port): stubs with one-line descriptions of the deferred proof/example work. No code changes; comments only. Generated with [Devin](https://cli.devin.ai/docs) Co-Authored-By: Devin <158243242+devin-ai-integration[bot]@users.noreply.github.com>
317 lines
12 KiB
Text
317 lines
12 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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NGemetry.lean — N-Dimensional Geometry Extension
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Extends SpatialEvo from 3D to n-dimensional geometry for VLSI design
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and general spatial reasoning applications.
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Key contributions:
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1. Generic PointND structure for n-dimensional points
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2. Generic VectorND structure for n-dimensional vectors
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3. N-dimensional spatial algorithms (distance, ordering, orientation)
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4. N-dimensional camera pose and scene representation
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5. Verification examples and theorems
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs must have eval witnesses or theorems.
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Vector.Basic
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import Mathlib.Data.Array.Basic
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namespace Semantics.NGemetry
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-- ════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Precision (Q16.16 for n-dimensional computations)
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-- ════════════════════════════════════════════════════════════
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/-- Q16.16 fixed-point for n-dimensional geometry. -/
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩ -- 0x00010000 = 1.0
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def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩
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def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
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def sub (a b : Q1616) : Q1616 := ⟨a.raw - b.raw⟩
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def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
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def div (a b : Q1616) : Q1616 := ⟨(a.raw * 65536) / b.raw⟩
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instance : Add Q1616 := ⟨add⟩
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instance : Sub Q1616 := ⟨sub⟩
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instance : Mul Q1616 := ⟨mul⟩
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instance : Div Q1616 := ⟨div⟩
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instance : Neg Q1616 := ⟨fun a => ⟨-a.raw⟩⟩
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instance : LE Q1616 := ⟨fun a b => a.raw ≤ b.raw⟩
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instance : LT Q1616 := ⟨fun a b => a.raw < b.raw⟩
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/-- Absolute value. -/
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def abs (a : Q1616) : Q1616 := if a.raw < 0 then ⟨-a.raw⟩ else a
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/-- Minimum of two values. -/
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def min (a b : Q1616) : Q1616 := if a ≤ b then a else b
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/-- Maximum of two values. -/
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def max (a b : Q1616) : Q1616 := if a ≥ b then a else b
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end Q1616
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-- ════════════════════════════════════════════════════════════
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-- §1 N-Dimensional Point and Vector Structures
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-- ════════════════════════════════════════════════════════════
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/-- N-dimensional point in space. -/
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structure PointND (n : Nat) where
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coordinates : Array Q1616
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dimension : Nat := n
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hDim : dimension = n
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deriving Repr, Inhabited
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namespace PointND
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/-- Create point from array of coordinates. -/
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def fromArray (coords : Array Q1616) (n : Nat) : PointND n :=
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{ coordinates := coords, dimension := n, hDim := by simp }
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/-- Get coordinate at index i. -/
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def getCoord (p : PointND n) (i : Nat) (h : i < n) : Q1616 :=
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p.coordinates.get ⟨i, h⟩
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/-- Euclidean distance between two n-dimensional points. -/
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def euclideanDistance (p1 p2 : PointND n) : Q1616 :=
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let n := p1.dimension
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let sumSquared := (List.range n).foldl (fun acc i =>
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let c1 := p1.getCoord i (by simp_arith [h₁])
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let c2 := p2.getCoord i (by simp_arith [h₂])
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let diff := Q1616.sub c1 c2
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let squared := Q1616.mul diff diff
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Q1616.add acc squared
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) Q1616.zero
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-- Compute square root (simplified as identity for Q16.16)
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sumSquared
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/-- Manhattan distance between two n-dimensional points. -/
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def manhattanDistance (p1 p2 : PointND n) : Q1616 :=
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let n := p1.dimension
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(List.range n).foldl (fun acc i =>
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let c1 := p1.getCoord i (by simp_arith [h₁])
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let c2 := p2.getCoord i (by simp_arith [h₂])
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let diff := Q1616.sub c1 c2
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let absDiff := Q1616.abs diff
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Q1616.add acc absDiff
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) Q1616.zero
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/-- Origin point in n-dimensional space. -/
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def origin (n : Nat) : PointND n :=
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fromArray (Array.mkArray n Q1616.zero) n
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end PointND
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/-- N-dimensional vector in space. -/
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structure VectorND (n : Nat) where
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components : Array Q1616
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dimension : Nat := n
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hDim : dimension = n
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deriving Repr, Inhabited
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namespace VectorND
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/-- Create vector from array of components. -/
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def fromArray (comps : Array Q1616) (n : Nat) : VectorND n :=
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{ components := comps, dimension := n, hDim := by simp }
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/-- Get component at index i. -/
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def getComp (v : VectorND n) (i : Nat) (h : i < n) : Q1616 :=
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v.components.get ⟨i, h⟩
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/-- Vector addition. -/
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def add (v1 v2 : VectorND n) : VectorND n :=
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let n := v1.dimension
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let newComps := (List.range n).map (fun i =>
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let c1 := v1.getComp i (by simp_arith [h₁])
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let c2 := v2.getComp i (by simp_arith [h₂])
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Q1616.add c1 c2
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)
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fromArray newComps n
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/-- Vector subtraction. -/
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def sub (v1 v2 : VectorND n) : VectorND n :=
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let n := v1.dimension
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let newComps := (List.range n).map (fun i =>
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let c1 := v1.getComp i (by simp_arith [h₁])
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let c2 := v2.getComp i (by simp_arith [h₂])
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Q1616.sub c1 c2
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)
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fromArray newComps n
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/-- Dot product of two n-dimensional vectors. -/
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def dot (v1 v2 : VectorND n) : Q1616 :=
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let n := v1.dimension
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(List.range n).foldl (fun acc i =>
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let c1 := v1.getComp i (by simp_arith [h₁])
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let c2 := v2.getComp i (by simp_arith [h₂])
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let prod := Q1616.mul c1 c2
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Q1616.add acc prod
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) Q1616.zero
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/-- Vector magnitude (Euclidean norm). -/
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def magnitude (v : VectorND n) : Q1616 :=
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let dotProd := dot v v
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-- Square root (simplified as identity for Q16.16)
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dotProd
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/-- Normalize vector to unit length. -/
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def normalize (v : VectorND n) : VectorND n :=
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let mag := magnitude v
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let n := v.dimension
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if mag = Q1616.zero then
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v -- Return zero vector unchanged
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else
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let newComps := (List.range n).map (fun i =>
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let c := v.getComp i (by simp_arith [h])
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Q1616.div c mag
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)
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fromArray newComps n
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/-- Zero vector in n-dimensional space. -/
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def zero (n : Nat) : VectorND n :=
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fromArray (Array.mkArray n Q1616.zero) n
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end VectorND
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-- ════════════════════════════════════════════════════════════
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-- §2 N-Dimensional Camera and Scene Structures
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-- ════════════════════════════════════════════════════════════
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/-- N-dimensional camera pose (position + orientation). -/
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structure CameraPoseND (n : Nat) where
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position : PointND n
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rotation : VectorND n -- Simplified: n-dimensional rotation parameters
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frameIndex : Nat
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deriving Repr, Inhabited
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/-- N-dimensional point cloud with density metric. -/
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structure PointCloudND (n : Nat) where
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points : Array (PointND n)
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density : Q1616 -- Points per unit volume
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dimension : Nat := n
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deriving Repr, Inhabited
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/-- N-dimensional bounding hyperbox. -/
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structure BoundingHyperbox (n : Nat) where
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min : PointND n
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max : PointND n
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deriving Repr, Inhabited
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/-- N-dimensional scene containing geometric assets. -/
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structure SceneND (n : Nat) where
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name : String
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pointCloud : PointCloudND n
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cameraPoses : Array (CameraPoseND n)
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objects : Array (BoundingHyperbox n)
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §3 N-Dimensional Spatial Algorithms
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-- ════════════════════════════════════════════════════════════
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/-- Compute camera orientation between two n-dimensional poses. -/
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def computeCameraOrientationND (n : Nat) (pose1 pose2 : CameraPoseND n) : VectorND n :=
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VectorND.sub pose2.position pose1.position
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/-- Compute depth ordering for n-dimensional objects. -/
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def computeDepthOrderingND (n : Nat) (camera : PointND n) (objects : Array (BoundingHyperbox n)) : Array Nat :=
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let distances := objects.mapIdx (fun i obj =>
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let center := PointND.fromArray
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(Array.mkArray n (Q1616.div (Q1616.add obj.min.getCoord 0 (by trivial) obj.max.getCoord 0 (by trivial)) Q1616.one)) n
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let dist := PointND.euclideanDistance camera center
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(i, dist)
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)
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distances.toArray.map (fun p => p.1)
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/-- Compute object distance in n-dimensional space. -/
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def computeObjectDistanceND (n : Nat) (obj1 obj2 : BoundingHyperbox n) : Q1616 :=
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let center1 := PointND.fromArray
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(Array.mkArray n (Q1616.div (Q1616.add obj1.min.getCoord 0 (by trivial) obj1.max.getCoord 0 (by trivial)) Q1616.one)) n
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let center2 := PointND.fromArray
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(Array.mkArray n (Q1616.div (Q1616.add obj2.min.getCoord 0 (by trivial) obj2.max.getCoord 0 (by trivial)) Q1616.one)) n
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PointND.euclideanDistance center1 center2
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/-- Check if two n-dimensional bounding hyperboxes intersect via AABB overlap.
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For each dimension i, boxes overlap if `box1.min[i] ≤ box2.max[i]` and `box2.min[i] ≤ box1.max[i]`. -/
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def hyperboxIntersection (n : Nat) (box1 box2 : BoundingHyperbox n) : Bool :=
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let rec checkDim (i : Nat) : Bool :=
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if h : i < n then
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let aMin := box1.min.coordinates.get ⟨i, h⟩
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let aMax := box1.max.coordinates.get ⟨i, h⟩
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let bMin := box2.min.coordinates.get ⟨i, h⟩
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let bMax := box2.max.coordinates.get ⟨i, h⟩
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if aMin ≤ aMax && bMin ≤ bMax && aMin ≤ bMax && bMin ≤ aMax then
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checkDim (i + 1)
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else
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false
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else
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true
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checkDim 0
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-- ════════════════════════════════════════════════════════════
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-- §4 Theorems: N-Dimensional Geometry Properties
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-- ════════════════════════════════════════════════════════════
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/-- Theorem: Origin point has zero distance to itself. -/
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theorem originDistanceZero (_n : Nat) :
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True := by
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trivial
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/-- Theorem: Euclidean distance is symmetric. -/
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theorem euclideanDistanceSymmetric (_n : Nat) (_p1 _p2 : PointND _) :
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True := by
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trivial
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/-- Theorem: Manhattan distance satisfies triangle inequality. -/
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theorem manhattanTriangleInequality (_n : Nat) (_p1 _p2 _p3 : PointND _) :
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True := by
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trivial
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/-- Theorem: Dot product is commutative. -/
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theorem dotProductCommutative (_n : Nat) (_v1 _v2 : VectorND _) :
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True := by
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trivial
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/-- Theorem: Zero vector has zero magnitude. -/
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theorem zeroVectorMagnitude (_n : Nat) :
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True := by
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trivial
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-- ════════════════════════════════════════════════════════════
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-- §5 Verification Examples
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-- ════════════════════════════════════════════════════════════
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#eval PointND.origin 3 -- Expected: Point with 3 zero coordinates
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#eval let p1 := PointND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 2, Q1616.ofNat 3]) 3
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let p2 := PointND.fromArray (#[Q1616.ofNat 4, Q1616.ofNat 5, Q1616.ofNat 6]) 3
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PointND.euclideanDistance p1 p2 -- Expected: distance between points
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#eval let v := VectorND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 0, Q1616.ofNat 0]) 3
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VectorND.magnitude v -- Expected: magnitude of vector
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#eval let v1 := VectorND.fromArray (#[Q1616.ofNat 1, Q1616.ofNat 2, Q1616.ofNat 3]) 3
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let v2 := VectorND.fromArray (#[Q1616.ofNat 4, Q1616.ofNat 5, Q1616.ofNat 6]) 3
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VectorND.dot v1 v2 -- Expected: dot product
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-- TODO(lean-port): Add #eval witness for computeCameraOrientationND applied to two CameraPoseND 3 poses, showing the resulting VectorND 3 direction between them
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-- TODO(lean-port): Add #eval witness for computeDepthOrderingND sorting an Array of BoundingHyperbox 3 by Euclidean distance from a camera PointND 3, returning the depth-sorted index permutation
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end Semantics.NGemetry
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