import Semantics.FixedPoint
import Semantics.Bind

namespace Semantics.YangMillsCompressionBounds

/-! ## Yang-Mills Compression Bounds

Formalization of compression bounds separating:
- Lossless compression (exact reconstruction)
- Lossy compression (acceptable precision loss)
- Precision-reduced compression (fixed-point conversion)
- Transmission avoidance (Layer 3 local computation)

Key invariant: compressionRatio ≠ transmissionAvoidance
Layer 3 can reduce when data is transmitted. It does not make the data smaller by itself.
-/

open Semantics.Q16_16

/-- Compression type classification. -/
inductive CompressionType where
  | lossless : CompressionType  -- Exact reconstruction
  | lossy : CompressionType  -- Acceptable precision loss
  | precisionReduced : CompressionType  -- Fixed-point conversion
  | transmissionAvoidance : CompressionType  -- Layer 3 local computation
deriving Repr, DecidableEq

/-- Compression bounds for each type. -/
structure CompressionBounds where
  compressionType : CompressionType
  minRatio : Q16_16  -- Minimum compression ratio (≥ 1)
  maxRatio : Q16_16  -- Maximum compression ratio (theoretical limit)
  requiresBenchmark : Bool  -- Requires empirical benchmark
  justification : String
deriving Repr

/-- Lossless compression bounds (Delta GCL). -/
-- Based on actual achievements: 92% on structured data, 99.9% on metadata
-- Field data less compressible: 2-5× realistic, 10-20× theoretical
def losslessBounds : CompressionBounds :=
  { compressionType := CompressionType.lossless
    minRatio := two  -- 2× (conservative)
    maxRatio := ofNat 20  -- 20× (theoretical upper bound for field data)
    requiresBenchmark := true
    justification := "Based on Delta GCL achievements (92% on structured data). Field data less compressible than metadata. Requires empirical zstd/ndzip comparison." }

/-- Lossy compression bounds (neural VAE). -/
-- Optional second stage with acceptable precision loss
def lossyBounds : CompressionBounds :=
  { compressionType := CompressionType.lossy
    minRatio := two  -- 2× (conservative)
    maxRatio := ofNat 20  -- 20× (depends on acceptable precision loss)
    requiresBenchmark := true
    justification := "Neural VAE compression with acceptable precision loss. Requires benchmark against precision requirements." }

/-- Precision-reduced compression bounds (fixed-point). -/
-- Float64 → Q16_16: 2× reduction in bit width
def precisionReducedBounds : CompressionBounds :=
  { compressionType := CompressionType.precisionReduced
    minRatio := two  -- 2× (Float64 → Q16_16)
    maxRatio := two  -- 2× (fixed, no variation)
    requiresBenchmark := false
    justification := "Fixed-point conversion: Float64 → Q16_16 is exactly 2× bit width reduction. No benchmark required." }

/-- Transmission avoidance bounds (Layer 3). -/
-- Layer 3 does NOT compress data - it avoids transmission during local computation
-- This is NOT a compression ratio, it's a transmission reduction factor
def transmissionAvoidanceBounds : CompressionBounds :=
  { compressionType := CompressionType.transmissionAvoidance
    minRatio := one  -- 1× (no compression)
    maxRatio := one  -- 1× (no compression)
    requiresBenchmark := false
    justification := "Layer 3 local computation avoids transmission but does NOT compress data. transmissionAvoidance ≠ compressionRatio." }

/-- Calculate effective network cost. -/
-- effective_network_cost = anchor_frequency × compressed_payload_size
structure NetworkCost where
  anchorFrequency : Nat  -- Number of anchors per unit time
  compressedPayloadSize : Nat  -- Size after compression
  effectiveCost : Nat  -- Total network cost
deriving Repr

/-- Calculate effective network cost.
--
-- Arithmetic sanity check:
-- effective_network_cost = anchor_frequency × compressed_payload_size.
--
-- Example:
-- 10 anchors × 1 MiB = 10 MiB.
--
-- Provenance note:
-- This is not a compression ratio. It is a scheduling/transmission-cost model.
--
-- 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 effectiveNetworkCost (anchorFreq : Nat) (payloadSize : Nat) : NetworkCost :=
  {
    anchorFrequency := anchorFreq
    compressedPayloadSize := payloadSize
    effectiveCost := anchorFreq * payloadSize
  }

#eval effectiveNetworkCost 10 1000000  -- 10 anchors × 1MB = 10MB effective cost

/-- Theorem: Lossless compression ratio ≥ 1. -/
theorem lossless_ratio_ge_one : losslessBounds.minRatio ≥ one := by
  native_decide

/-- Theorem: Precision reduction is exactly 2×. -/
theorem precision_reduction_exact_two : precisionReducedBounds.minRatio = two ∧ precisionReducedBounds.maxRatio = two := by
  native_decide

/-- Theorem: Transmission avoidance does NOT compress data. -/
theorem transmission_avoidance_no_compression : transmissionAvoidanceBounds.minRatio = one ∧ transmissionAvoidanceBounds.maxRatio = one := by
  native_decide

/-- Theorem: compressionRatio ≠ transmissionAvoidance. -/
theorem compression_not_transmission_avoidance :
    losslessBounds.compressionType ≠ CompressionType.transmissionAvoidance := by
  native_decide

/-- Theorem: Effective network cost is anchor frequency × payload size. -/
theorem effective_cost_formula (anchorFreq : Nat) (payloadSize : Nat) :
    (effectiveNetworkCost anchorFreq payloadSize).effectiveCost = anchorFreq * payloadSize := by
  rfl

end Semantics.YangMillsCompressionBounds
