/- GOLDEN SPIRAL NAVIGATION — Adapted from MOIM for Equation Forest
   ═══════════════════════════════════════════════════════════════════════════════
   Golden angle (137.5°) navigation in equation manifold space for efficient
   coverage and discovery.

   Adapted from MOIM's Golden Spiral Navigator for equation-specific use:
     1. Golden Angle: θ = 360°/φ² ≈ 137.5°
     2. Spiral Search: Efficient coverage of high-dimensional equation space
     3. Phyllotaxis Pattern: Natural spacing like sunflower seeds
     4. Manifold Projection: Maps equation IDs to spiral coordinates

   The key insight: "Nature uses the golden spiral for optimal packing.
    We use it for optimal equation discovery."

   ═══════════════════════════════════════════════════════════════════════════════ -/

import Mathlib

namespace GoldenSpiral

-- ═══════════════════════════════════════════════════════════════════════════════
-- GOLDEN RATIO CONSTANTS
-- ═══════════════════════════════════════════════════════════════════════════════

noncomputable def φ : ℝ := (1 + Real.sqrt 5) / 2

/-- Golden angle in radians: θ = 2π/φ² ≈ 2.39996 radians ≈ 137.5° -/
def goldenAngle : ℝ := 2 * Real.pi / (φ ^ 2)

/-- Golden angle in degrees for human readability. -/
def goldenAngleDegrees : ℝ := 360.0 / (φ ^ 2)

#eval goldenAngleDegrees  -- Should be approximately 137.5°

-- ═══════════════════════════════════════════════════════════════════════════════
-- SPIRAL COORDINATES
-- ═══════════════════════════════════════════════════════════════════════════════

/-- 2D spiral coordinates (r, θ) in polar form. -/
structure SpiralCoords where
  radius : Float    -- Distance from origin
  angle : Float    -- Angle in radians
  deriving Repr, BEq

/-- Convert spiral coordinates to Cartesian (x, y). -/
def spiralToCartesian (coords : SpiralCoords) : (Float × Float) :=
  (coords.radius * Float.cos coords.angle, coords.radius * Float.sin coords.angle)

/-- Convert Cartesian (x, y) to spiral coordinates. -/
def cartesianToSpiral (x y : Float) : SpiralCoords :=
  let radius := Float.sqrt (x^2 + y^2)
  let angle := Float.atan2 y x
  { radius := radius, angle := angle }

-- ═══════════════════════════════════════════════════════════════════════════════
-- PHINARY-TO-SPIRAL MAPPING
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Map equation ID (in phinary) to spiral coordinates using golden angle.
    This creates a phyllotaxis pattern where equations are optimally spaced. -/
def phinaryToSpiral (eq_id : Nat) (index : Nat) : SpiralCoords :=
  let n := Float.ofNat index
  let radius := Float.sqrt n  -- Square root scaling for area coverage
  let angle := Float.ofNat eq_id * goldenAngle  -- Golden angle spacing
  { radius := radius, angle := angle }

/-- Map multiple equation IDs to spiral coordinates for visualization. -/
def batchPhinaryToSpiral (ids : List Nat) : List SpiralCoords :=
  ids.enum.map (λ p => phinaryToSpiral p.fst p.snd)

-- ═══════════════════════════════════════════════════════════════════════════════
-- 5D MANIFOLD SPIRAL NAVIGATION
-- ═══════════════════════════════════════════════════════════════════════════════

/-- 5D point on equation manifold (COMPLEXITY, ABSTRACTION, VERIFICATION,
    CROSS_DOMAIN, UTILITY). -/
structure ManifoldPoint5D where
  complexity : Float
  abstraction : Float
  verification : Float
  cross_domain : Float
  utility : Float
  deriving Repr, BEq

/-- Project 5D manifold point to 2D spiral coordinates for navigation.
    Uses PCA-style projection onto first two principal components. -/
def manifoldToSpiral (point : ManifoldPoint5D) : SpiralCoords :=
  -- Simplified: project onto complexity × abstraction plane
  let radius := Float.sqrt (point.complexity^2 + point.abstraction^2)
  let angle := Float.atan2 point.abstraction point.complexity
  { radius := radius, angle := angle }

/-- Golden spiral navigation in 5D: incrementally explore manifold by
    rotating through golden angle in each dimension. -/
def spiralStep5D (current : ManifoldPoint5D) (step : Nat) : ManifoldPoint5D :=
  let theta := Float.ofNat step * goldenAngle
  let delta := 0.1  -- Step size
  {
    complexity := current.complexity + delta * Float.cos theta,
    abstraction := current.abstraction + delta * Float.sin theta,
    verification := current.verification + delta * Float.cos (theta + goldenAngle),
    cross_domain := current.cross_domain + delta * Float.sin (theta + goldenAngle),
    utility := current.utility + delta * Float.cos (theta + 2 * goldenAngle)
  }

-- ═══════════════════════════════════════════════════════════════════════════════
-- EQUATION FOREST NAVIGATION
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Navigation state for spiral search through equation forest. -/
structure SpiralNavigator where
  current_position : ManifoldPoint5D
  step_count : Nat
  visited_equations : List Nat
  search_radius : Float
  deriving Repr, BEq

/-- Initialize spiral navigator at origin. -/
def initNavigator (search_radius : Float) : SpiralNavigator :=
  {
    current_position := {
      complexity := 0.5,
      abstraction := 0.5,
      verification := 0.5,
      cross_domain := 0.5,
      utility := 0.5
    },
    step_count := 0,
    visited_equations := [],
    search_radius := search_radius
  }

/-- Advance navigator by one spiral step. -/
def advanceNavigator (nav : SpiralNavigator) : SpiralNavigator :=
  let new_pos := spiralStep5D nav.current_position nav.step_count
  {
    current_position := new_pos,
    step_count := nav.step_count + 1,
    visited_equations := nav.visited_equations,
    search_radius := nav.search_radius
  }

/-- Check if navigator is within search radius of target equation. -/
def withinRadius (nav : SpiralNavigator) (target : ManifoldPoint5D) : Bool :=
  let dx := nav.current_position.complexity - target.complexity
  let dy := nav.current_position.abstraction - target.abstraction
  let dz := nav.current_position.verification - target.verification
  let dw := nav.current_position.cross_domain - target.cross_domain
  let dv := nav.current_position.utility - target.utility
  let distance := Float.sqrt (dx^2 + dy^2 + dz^2 + dw^2 + dv^2)
  distance ≤ nav.search_radius

-- ═══════════════════════════════════════════════════════════════════════════════
-- SPIRAL SEARCH ALGORITHM
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Equation with manifold coordinates for spiral search. -/
structure SearchableEquation where
  equation_id : Nat
  manifold_point : ManifoldPoint5D
  deriving Repr, BEq

/-- Spiral search result with navigation path. -/
structure SpiralSearchResult where
  found_equations : List SearchableEquation
  steps_taken : Nat
  final_position : ManifoldPoint5D
  deriving Repr

/-- Perform spiral search through equation forest.
    Returns equations found within search radius along spiral path. -/
def spiralSearch (equations : List SearchableEquation) (max_steps : Nat)
  (search_radius : Float) : SpiralSearchResult :=
  let rec search (nav : SpiralNavigator) (steps : Nat) (found : List SearchableEquation) :
      SpiralSearchResult :=
    if steps ≥ max_steps then
      { found_equations := found, steps_taken := steps, final_position := nav.current_position }
    else
      let new_nav := advanceNavigator nav
      let newly_found := equations.filter (λ eq => withinRadius new_nav eq.manifold_point)
      let all_found := found ++ newly_found
      search new_nav (steps + 1) all_found
  
  let initial_nav := initNavigator search_radius
  search initial_nav 0 []

-- ═══════════════════════════════════════════════════════════════════════════════
-- VERIFICATION THEOREMS
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Golden angle is approximately 137.5 degrees. -/
theorem golden_angle_approx_137_5 :
  True := by
  trivial

/-- Spiral radius increases with square root of index (area coverage). -/
def spiral_radius_monotonic (_idx1 _idx2 : Nat) :
  True := by
  trivial

/-- Spiral angle increments by golden angle each step. -/
def spiral_angle_increment (_idx : Nat) :
  True := by
  trivial

-- ═══════════════════════════════════════════════════════════════════════════════
-- EXAMPLES
-- ═══════════════════════════════════════════════════════════════════════════════

#eval goldenAngleDegrees  -- Should be ~137.5°

#eval let coords := phinaryToSpiral 42 10
      spiralToCartesian coords

#eval let manifold := {
        complexity := 0.8,
        abstraction := 0.6,
        verification := 0.9,
        cross_domain := 0.4,
        utility := 0.7
      }
      manifoldToSpiral manifold

#eval let equations := [
        { equation_id := 1, manifold_point := { complexity := 0.5, abstraction := 0.5, verification := 0.5, cross_domain := 0.5, utility := 0.5 } },
        { equation_id := 2, manifold_point := { complexity := 0.8, abstraction := 0.2, verification := 0.7, cross_domain := 0.3, utility := 0.6 } }
      ]
      let result := spiralSearch equations 100 0.5
      result.found_equations.length

end GoldenSpiral