import Semantics.FixedPoint
import Semantics.Bind
import Mathlib.Tactic

namespace Semantics.YangMillsLatticeSizing

/-! ## Yang-Mills Lattice Sizing Models

Formalization of lattice site count and storage models for small lattice experiments.
This is NOT a Yang-Mills proof stack - it is a lattice-gauge / compression / verification sandbox.

Key invariants:
- Small lattice experiments only (toy models)
- Storage models for 8-real/site vs full SU(3) link model
- No claim to 64⁴ production feasibility
-/

open Semantics.Q16_16

/-- Lattice sizing parameters for toy models. -/
structure ToyLatticeConfig where
  L : Nat  -- Linear dimension (small for toy models: 4-16)
  gaugeGroup : Nat  -- Gauge group (2 for SU(2), 3 for SU(3))
  storageModel : String  -- "8-real/site" or "SU(3) link model"
  bytesPerReal : Nat  -- Bytes per real number (4 for Q16_16, 8 for Float64)
deriving Repr

/-- Default toy lattice configuration (8-real/site model). -/
def defaultToyLattice : ToyLatticeConfig :=
  { L := 8  -- Small for toy experiments
    gaugeGroup := 3
    storageModel := "8-real/site"
    bytesPerReal := 4 }  -- Q16_16 fixed-point

/-- Calculate total lattice sites: N = L⁴.
--
-- Arithmetic sanity check:
-- 8^4 = 4096 and 16^4 = 65536.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def latticeSites (config : ToyLatticeConfig) : Nat :=
  config.L ^ 4

/-- Calculate raw storage size based on model. -/
def rawStorageSize (config : ToyLatticeConfig) : Nat :=
  match config.storageModel with
  | "8-real/site" => latticeSites config * 8 * config.bytesPerReal
  | "SU(3) link model" => latticeSites config * 4 * 2 * config.bytesPerReal  -- 4 complex × 2 (real/imag)
  | _ => latticeSites config * 8 * config.bytesPerReal  -- Default to 8-real/site

/-- Convert bytes to megabytes.
--
-- Arithmetic sanity check:
-- 1048576 bytes / 1048576 = 1 MB.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def bytesToMegabytes (bytes : Nat) : Q16_16 :=
  satFromNat (bytes * scale / (1024 * 1024))

/-- Calculate raw storage size in megabytes. -/
def rawStorageSizeMB (config : ToyLatticeConfig) : Q16_16 :=
  bytesToMegabytes (rawStorageSize config)

#eval rawStorageSizeMB defaultToyLattice  -- Expected: ~1 MB for 8⁴ lattice with Q16_16

/-- Theorem: Lattice sites are positive when the linear dimension is positive. -/
theorem latticeSites_positive (config : ToyLatticeConfig) (hL : config.L > 0) :
    latticeSites config > 0 := by
  simp [latticeSites]
  exact Nat.pow_pos hL

/-- Theorem: Raw storage size is positive for nonempty lattices and nonzero reals. -/
theorem rawStorageSize_positive (config : ToyLatticeConfig)
    (hL : config.L > 0) (hBytes : config.bytesPerReal > 0) :
    rawStorageSize config > 0 := by
  unfold rawStorageSize
  split
  · exact Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) hBytes
  · exact Nat.mul_pos
      (Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) (by decide))
      hBytes
  · exact Nat.mul_pos (Nat.mul_pos (latticeSites_positive config hL) (by decide)) hBytes

/-- Theorem: 8-real/site model uses 8 reals per site. -/
theorem eightRealModel_eightReals (config : ToyLatticeConfig) :
    config.storageModel = "8-real/site" →
      rawStorageSize config = latticeSites config * 8 * config.bytesPerReal := by
  intro hModel
  simp [rawStorageSize, hModel]

/-- Theorem: SU(3) link model uses 8 reals per site (4 complex × 2). -/
def su3LinkModel_eightReals (config : ToyLatticeConfig) :
    config.storageModel = "SU(3) link model" →
      rawStorageSize config = latticeSites config * 8 * config.bytesPerReal := by
  intro hModel
  simp [rawStorageSize, hModel]
  ring_nf
  exact Or.inl trivial

/-- Toy lattice feasibility check. -/
structure FeasibilityCheck where
  feasible : Bool
  reason : String
  maxL : Nat  -- Maximum feasible L for given constraints
  maxMB : Q16_16  -- Maximum storage in MB
deriving Repr

/-- Check if lattice is feasible for toy experiments. -/
def feasibilityCheck (config : ToyLatticeConfig) (maxStorageMB : Q16_16) : FeasibilityCheck :=
  let sizeMB := rawStorageSizeMB config
  let feasible := sizeMB ≤ maxStorageMB
  let maxL := if config.L ≤ 16 then 16 else config.L  -- Conservative max for toy models
  {
    feasible := feasible
    reason := if feasible then "Lattice fits within storage constraints" else "Lattice exceeds storage constraints"
    maxL := maxL
    maxMB := maxStorageMB
  }

#eval feasibilityCheck defaultToyLattice (ofNat 16)  -- Check if 16MB feasible (should be feasible)

end Semantics.YangMillsLatticeSizing