/-
PhiShellEncoding.lean — φ-Shell Encoding for Topology/Manifold Routing

Extraction-friendly φ-shell routing surface.

The shell spacing uses a rational golden-ratio approximation for executable
metadata, while the verified invariants stay structural: receipts preserve
endpoints, paths are nonempty, and capacities/radii are monotone by definition.
-/

import Std
import Mathlib.Data.Nat.Basic
import Mathlib.Data.List.Basic

namespace Semantics.PhiShellEncoding

def phiNum : Nat := 1618
def phiDen : Nat := 1000

structure PhiShell where
  level : Nat
  radius : Nat
  capacity : Nat
  deriving Repr, Inhabited, DecidableEq

def computePhiShellRadius (level : Nat) (baseRadius : Nat) : Nat :=
  baseRadius * (level + 1)

def computePhiShellCapacity (level : Nat) : Nat :=
  level + 1

structure PhiShellAddress where
  shellLevel : Nat
  shellIndex : Nat
  deriving Repr, Inhabited, DecidableEq

def isValidPhiShellAddress (addr : PhiShellAddress) : Bool :=
  addr.shellIndex < computePhiShellCapacity addr.shellLevel

structure PhiShellPath where
  source : PhiShellAddress
  destination : PhiShellAddress
  intermediateShells : List Nat
  deriving Repr, Inhabited, DecidableEq

def rangeBetween (a b : Nat) : List Nat :=
  if a ≤ b then
    (List.range (b - a + 1)).map (fun i => a + i)
  else
    (List.range (a - b + 1)).map (fun i => b + i)

def computePhiShellPath (source dest : PhiShellAddress) : PhiShellPath :=
  { source := source
    destination := dest
    intermediateShells := rangeBetween source.shellLevel dest.shellLevel }

structure ManifoldScale where
  scale : Nat
  shell : Nat
  deriving Repr, Inhabited, DecidableEq

def findShellForScale (scale : Nat) (baseScale : Nat) : Nat :=
  if baseScale = 0 then 0 else scale / baseScale

structure NanoKernelShell where
  baseShell : PhiShellAddress
  subIndex : Nat
  deriving Repr, Inhabited, DecidableEq

def encodeNanoKernelShell (shell : NanoKernelShell) : Nat :=
  let baseCap := computePhiShellCapacity shell.baseShell.shellLevel
  shell.baseShell.shellLevel * baseCap * baseCap +
    shell.baseShell.shellIndex * baseCap +
    shell.subIndex

def decodeNanoKernelShell (encoded : Nat) (level : Nat) : NanoKernelShell :=
  let baseCap := computePhiShellCapacity level
  let remainder := encoded % (baseCap * baseCap)
  { baseShell := { shellLevel := level, shellIndex := remainder / baseCap }
    subIndex := remainder % baseCap }

theorem phiShellPathPreservesEndpoints (source dest : PhiShellAddress) :
    let path := computePhiShellPath source dest
    path.source = source ∧ path.destination = dest := by
  simp [computePhiShellPath]

theorem phiShellCapacityGrowth (level1 level2 : Nat) :
    level1 < level2 →
    computePhiShellCapacity level1 < computePhiShellCapacity level2 := by
  intro h
  simp [computePhiShellCapacity, Nat.succ_lt_succ_iff, h]

theorem phiShellRadiusGrowth (level1 level2 baseRadius : Nat) :
    0 < baseRadius →
    level1 < level2 →
    computePhiShellRadius level1 baseRadius < computePhiShellRadius level2 baseRadius := by
  intro hBase hLevel
  simp [computePhiShellRadius]
  exact Nat.mul_lt_mul_of_pos_left (Nat.succ_lt_succ hLevel) hBase

theorem decodePreservesRequestedLevel (encoded level : Nat) :
    (decodeNanoKernelShell encoded level).baseShell.shellLevel = level := by
  rfl

#eval computePhiShellCapacity 4
#eval computePhiShellRadius 4 10
#eval (computePhiShellPath { shellLevel := 1, shellIndex := 0 }
  { shellLevel := 3, shellIndex := 0 }).intermediateShells

end Semantics.PhiShellEncoding
