import Semantics.FixedPoint
import Semantics.Bind

namespace Semantics.Layer3TransmissionModel

/-! ## Layer 3 Transmission Model

Formal proof that "0 bytes during local compute" is NOT compression.

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

Corrected architecture equation:
- effective_network_cost = anchor_frequency × compressed_payload_size
- NOT: raw_payload_size / 0 = ∞
-/

open Semantics.Q16_16

/-- Transmission event type. -/
inductive TransmissionEvent where
  | localComputation : TransmissionEvent  -- Compute locally, no transmission
  | anchorTransmission : Nat → TransmissionEvent  -- Transmit anchor with size
deriving Repr

/-- Transmission cost model. -/
structure TransmissionCost where
  eventType : TransmissionEvent
  dataSize : Nat  -- Size of data involved
  transmittedBytes : Nat  -- Bytes actually transmitted
  transmissionAvoidance : Bool  -- Whether transmission was avoided
deriving Repr

/-- Calculate transmission cost for local computation. -/
def localComputationCost (dataSize : Nat) : TransmissionCost :=
  {
    eventType := TransmissionEvent.localComputation
    dataSize := dataSize
    transmittedBytes := 0
    transmissionAvoidance := true
  }

/-- Calculate transmission cost for anchor transmission. -/
def anchorTransmissionCost (anchorSize : Nat) : TransmissionCost :=
  {
    eventType := TransmissionEvent.anchorTransmission anchorSize
    dataSize := anchorSize
    transmittedBytes := anchorSize
    transmissionAvoidance := false
  }

/-- Compression ratio calculation.
-- compressionRatio = originalSize / compressedSize
--
-- Arithmetic sanity check:
-- 1000 / 100 = 10× compression ratio.
--
-- 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 compressionRatio (originalSize : Nat) (compressedSize : Nat) : Q16_16 :=
  ofNat originalSize / ofNat compressedSize

/-- Theorem: Local computation has 0 transmitted bytes. -/
theorem local_computation_zero_transmitted (dataSize : Nat) :
    (localComputationCost dataSize).transmittedBytes = 0 := by
  rfl

/-- Theorem: Local computation does NOT change data size. -/
theorem local_computation_no_size_change (dataSize : Nat) :
    (localComputationCost dataSize).dataSize = dataSize := by
  rfl

/-- Theorem: Local computation is transmission avoidance, not size reduction. -/
theorem local_computation_not_compression (dataSize : Nat) :
    (localComputationCost dataSize).transmissionAvoidance = true ∧
      (localComputationCost dataSize).dataSize = dataSize := by
  exact ⟨rfl, rfl⟩

/-- Theorem: Transmission avoidance records no local payload shrink. -/
theorem transmission_avoidance_not_compression (dataSize : Nat) :
    (localComputationCost dataSize).transmissionAvoidance = true →
      (localComputationCost dataSize).dataSize = dataSize := by
  intro _hAvoided
  rfl

/-- Theorem: Effective network cost is anchor frequency × payload size. -/
-- This is the CORRECTED architecture equation
structure EffectiveNetworkCost 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 (corrected formula).
--
-- 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 calculateEffectiveCost (anchorFreq : Nat) (payloadSize : Nat) : EffectiveNetworkCost :=
  {
    anchorFrequency := anchorFreq
    compressedPayloadSize := payloadSize
    effectiveCost := anchorFreq * payloadSize
  }

/-- Theorem: Effective cost is NOT infinite. -/
theorem effective_cost_not_infinite (anchorFreq : Nat) (payloadSize : Nat)
    (hAnchor : anchorFreq > 0) (hPayload : payloadSize > 0) :
    (calculateEffectiveCost anchorFreq payloadSize).effectiveCost ≠ 0 := by
  change anchorFreq * payloadSize ≠ 0
  exact Nat.mul_ne_zero (Nat.ne_of_gt hAnchor) (Nat.ne_of_gt hPayload)

/-- Theorem: Effective cost formula is correct. -/
theorem effective_cost_formula_correct (anchorFreq : Nat) (payloadSize : Nat) :
    (calculateEffectiveCost anchorFreq payloadSize).effectiveCost = anchorFreq * payloadSize := by
  rfl

/-- In the Q16.16 compression model, a zero denominator is represented by the
    explicit `infinity` sentinel rather than by claiming a Nat-level infinity. -/
theorem compression_ratio_zero_denominator_is_infinity (n : Nat) :
    compressionRatio n 0 = infinity := by
  unfold compressionRatio
  change Semantics.Q16_16.div (ofNat n) (ofNat 0) = infinity
  simp [Semantics.Q16_16.div, ofNat, infinity]

end Semantics.Layer3TransmissionModel
