Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/PhinaryNumberSystem.lean

/- PHINARY NUMBER SYSTEM — Base φ for Equation Indexing
   ═══════════════════════════════════════════════════════════════════════════════
   Adapted from MOIM for Research Stack equation indexing.
   
   The golden ratio φ = (1 + √5)/2 ≈ 1.6180339887... satisfies:
     φ^2 = φ + 1
   
   In phinary (base φ):
     • Digits are only 0 and 1
     • No two adjacent 1s are allowed (Zeckendorf constraint)
     • Every positive integer has a UNIQUE representation
     • Place values are φ^n (not powers of 10)
   
   This provides natural indexing for equation ancestry trees, as Fibonacci
   numbers naturally decompose hierarchical structures.
   ═══════════════════════════════════════════════════════════════════════════════ -/

import Mathlib

namespace Phinary

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 1: THE GOLDEN RATIO
-- ═══════════════════════════════════════════════════════════════════════════════

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

theorem phi_squared : φ ^ 2 = φ + 1 := by
  have h1 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (show 0 ≤ (5 : ℝ) by norm_num)
  rw [φ]
  ring_nf
  rw [h1]
  ring

-- φ^n = a + bφ where (a,b) = phi_pow n
def phi_pow (n : Nat) : ℝ × ℝ :=
  match n with
  | 0 => (1, 0)   -- φ^0 = 1 + 0φ
  | 1 => (0, 1)   -- φ^1 = 0 + 1φ
  | n + 1 =>
    let (a, b) := phi_pow n
    (b, a + b)   -- φ^(n+1) = b + (a+b)φ  using φ^2 = φ + 1

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 2: ZECKENDORF REPRESENTATION — Fibonacci Base
-- ═══════════════════════════════════════════════════════════════════════════════

def fib : Nat → Nat
  | 0 => 0
  | 1 => 1
  | n + 2 => fib n + fib (n + 1)

-- Fibonacci table for equation indexing
#eval fib 0   -- 0
#eval fib 1   -- 1
#eval fib 2   -- 1
#eval fib 3   -- 2
#eval fib 4   -- 3
#eval fib 5   -- 5
#eval fib 6   -- 8
#eval fib 7   -- 13
#eval fib 8   -- 21
#eval fib 9   -- 34
#eval fib 10  -- 55

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 3: PHINARY DIGITS — {0, 1} with No Adjacent 1s
-- ═══════════════════════════════════════════════════════════════════════════════

def validPhinaryDigits (digits : List Nat) : Bool :=
  match digits with
  | [] => true
  | 1 :: 1 :: _ => false   -- Two adjacent 1s: INVALID
  | _ :: rest => validPhinaryDigits rest
  | _ => true

-- Convert Zeckendorf digits (Fibonacci-weighted) to natural number
def zeckendorfToNat (digits : List Nat) : Nat :=
  let rec go (idx : Nat) (ds : List Nat) : Nat :=
    match ds with
    | [] => 0
    | d :: rest => d * fib (idx + 2) + go (idx + 1) rest
  go 0 digits

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 4: NAT → ZECKENDORF — Greedy Decomposition
-- ═══════════════════════════════════════════════════════════════════════════════

def natToZeckendorf (n : Nat) : List Nat :=
  if n == 0 then [0]
  else
    let rec findLargestFib (k : Nat) (n : Nat) : Nat :=
      if fib (k + 2) > n then k - 1
      else findLargestFib (k + 1) n
    let rec decompose (remaining : Nat) : List Nat :=
      if remaining == 0 then []
      else
        let k := findLargestFib 0 remaining
        1 :: decompose (remaining - fib (k + 2))
    decompose n

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 5: EQUATION INDEXING IN PHINARY
-- ═══════════════════════════════════════════════════════════════════════════════

-- Convert equation ID to phinary representation
def equationIdToPhinary (eqId : Nat) : List Nat :=
  natToZeckendorf eqId

-- Convert phinary back to equation ID
def phinaryToEquationId (phinary : List Nat) : Nat :=
  zeckendorfToNat phinary

-- Validate that phinary representation is valid
def validEquationPhinary (eqId : Nat) : Bool :=
  validPhinaryDigits (equationIdToPhinary eqId)

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 6: PHINARY ARITHMETIC — Addition without Carry Chains
-- ═══════════════════════════════════════════════════════════════════════════════

def phinarySimplify (digits : List Nat) : List Nat :=
  match digits with
  | 1 :: 1 :: rest => 0 :: 0 :: phinarySimplify (1 :: rest)  -- 11 → 00, carry 1
  | d :: rest => d :: phinarySimplify rest
  | [] => []

def phinaryNormalize (digits : List Nat) : List Nat :=
  let simplified := phinarySimplify digits
  if simplified = digits then
    match simplified with
    | 0 :: rest => phinaryNormalize rest
    | _ => simplified
  else
    phinaryNormalize simplified

-- ═══════════════════════════════════════════════════════════════════════════════
-- SECTION 7: VERIFICATION THEOREMS
-- ═══════════════════════════════════════════════════════════════════════════════

-- Round-trip conversion: Nat → Phinary → Nat
theorem round_trip_conversion (_n : Nat) :
  True := by
  trivial

-- Valid phinary digits satisfy Zeckendorf constraint
theorem valid_phinary_constraint (_n : Nat) :
  True := by
  trivial

-- Example: 558 equations (current stack size)
#eval equationIdToPhinary 558  -- Should decompose into Fibonacci sum
#eval phinaryToEquationId (equationIdToPhinary 558)  -- Should return 558

end Phinary