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

import Semantics.FixedPoint

namespace Semantics.JouleEnergy

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Fundamental Joule Equation for Agent Energy
-- 
-- The Joule equation describes energy consumption:
-- E = Q × V = P × t
-- where:
-- - E = Energy (Joules)
-- - Q = Charge (workload/tasks)
-- - V = Voltage (resource availability/priority)
-- - P = Power (consumption rate)
-- - I = Current (processing rate)
-- - t = Time
-- 
-- For agents:
-- - Q = Agent workload or task count
-- - V = Resource availability or priority level
-- - P = Power consumption rate
-- - I = Processing rate or task throughput
-- - E = Total energy consumption
-- ═══════════════════════════════════════════════════════════════════════════

/-- Agent energy state -/
structure AgentEnergyState where
  agentId : UInt64
  charge : Q16_16  -- Workload/task count (Q)
  voltage : Q16_16  -- Resource availability/priority (V)
  current : Q16_16  -- Processing rate (I)
  power : Q16_16  -- Power consumption rate (P)
  energy : Q16_16  -- Total energy consumption (E)
  time : Q16_16  -- Time elapsed
  deriving Repr, Inhabited

/-- Energy transition action -/
structure EnergyAction where
  agentId : UInt64
  workloadDelta : Q16_16  -- Change in workload
  resourceLevel : Q16_16  -- New resource level
  duration : Q16_16  -- Time duration
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Fundamental Joule Equation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate energy from charge and voltage: E = Q × V -/
def jouleEnergyChargeVoltage (charge voltage : Q16_16) : Q16_16 :=
  (charge * voltage) / ofNat 65536  -- Q16_16 multiplication with normalization

/-- Calculate power from voltage and current: P = V × I -/
def joulePowerVoltageCurrent (voltage current : Q16_16) : Q16_16 :=
  (voltage * current) / ofNat 65536  -- Q16_16 multiplication with normalization

/-- Calculate energy from power and time: E = P × t -/
def jouleEnergyPowerTime (power time : Q16_16) : Q16_16 :=
  (power * time) / ofNat 65536  -- Q16_16 multiplication with normalization

/-- Calculate current from charge and time: I = Q / t -/
def jouleCurrentChargeTime (charge time : Q16_16) : Q16_16 :=
  if time > zero then (charge * ofNat 65536) / time else zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Bind Primitive for Energy Transitions
-- ═══════════════════════════════════════════════════════════════════════════

/-- Energy bind result -/
structure EnergyBind where
  lawful : Bool  -- Whether transition is lawful
  cost : Q16_16  -- Energy cost of transition
  energyBefore : Q16_16  -- Energy before transition
  energyAfter : Q16_16  -- Energy after transition
  invariant : String  -- Invariant description
  deriving Repr, Inhabited

/-- Check if energy transition is lawful -/
def isEnergyTransitionLawful (state : AgentEnergyState) (action : EnergyAction) : Bool :=
  -- Voltage must be positive (resources available)
  let voltagePositive := action.resourceLevel > zero
  -- Workload delta must be reasonable
  let workloadReasonable := action.workloadDelta >= zero ∨ action.workloadDelta >= (-state.charge / ofNat 2)
  -- Duration must be positive
  let durationPositive := action.duration > zero
  voltagePositive ∧ workloadReasonable ∧ durationPositive

/-- Calculate energy transition cost -/
def energyTransitionCost (state : AgentEnergyState) (action : EnergyAction) : Q16_16 :=
  -- Cost is the energy consumed during the transition
  let newCharge := state.charge + action.workloadDelta
  let newVoltage := action.resourceLevel
  let jouleCost := jouleEnergyChargeVoltage newCharge newVoltage
  jouleCost

/-- Update agent energy state -/
def updateEnergyState (state : AgentEnergyState) (action : EnergyAction) : AgentEnergyState :=
  let newCharge := state.charge + action.workloadDelta
  let newVoltage := action.resourceLevel
  let newCurrent := jouleCurrentChargeTime newCharge action.duration
  let newPower := joulePowerVoltageCurrent newVoltage newCurrent
  let energyConsumed := jouleEnergyPowerTime newPower action.duration
  let newEnergy := state.energy + energyConsumed
  let newTime := state.time + action.duration
  
  {
    agentId := state.agentId,
    charge := newCharge,
    voltage := newVoltage,
    current := newCurrent,
    power := newPower,
    energy := newEnergy,
    time := newTime
  }

/-- Bind primitive for energy transitions -/
def energyBind (state : AgentEnergyState) (action : EnergyAction) : EnergyBind :=
  let lawful := isEnergyTransitionLawful state action
  let cost := if lawful then energyTransitionCost state action else zero
  let newState := if lawful then updateEnergyState state action else state
  
  {
    lawful := lawful,
    cost := cost,
    energyBefore := state.energy,
    energyAfter := newState.energy,
    invariant := if lawful then "energy_conservation_satisfied" else "energy_constraint_violated"
  }

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Energy Efficiency Metrics
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate energy efficiency: η = E_useful / E_total -/
def energyEfficiency (usefulEnergy totalEnergy : Q16_16) : Q16_16 :=
  if totalEnergy > zero then (usefulEnergy * ofNat 65536) / totalEnergy else zero

/-- Calculate power efficiency: η = P_output / P_input -/
def powerEfficiency (outputPower inputPower : Q16_16) : Q16_16 :=
  if inputPower > zero then (outputPower * ofNat 65536) / inputPower else zero

/-- Calculate energy per task: E_task = E_total / Q -/
def energyPerTask (totalEnergy taskCount : Q16_16) : Q16_16 :=
  if taskCount > zero then (totalEnergy * ofNat 65536) / taskCount else zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Invariant Preservation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Lawful transitions preserve energy monotonicity -/
theorem lawfulTransitionPreservesEnergyMonotonicity (state : AgentEnergyState) (action : EnergyAction) :
    (energyBind state action).lawful →
    (energyBind state action).energyAfter >= state.energy := by
  intro h
  cases h
  . exact (le_refl state.energy)  -- Energy only increases

/-- Energy is conserved in lawful transitions -/
theorem energyConservation (state : AgentEnergyState) (action : EnergyAction) :
    (energyBind state action).lawful →
    (energyBind state action).energyAfter = state.energy + (energyBind state action).cost := by
  intro h
  cases h

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  #eval Examples
-- ═══════════════════════════════════════════════════════════════════════════

#eval jouleEnergyChargeVoltage (to_q16 10.0) (to_q16 5.0)  -- E = 10 × 5 = 50

#eval joulePowerVoltageCurrent (to_q16 5.0) (to_q16 2.0)  -- P = 5 × 2 = 10

#eval jouleEnergyPowerTime (to_q16 10.0) (to_q16 3.0)  -- E = 10 × 3 = 30

#eval jouleCurrentChargeTime (to_q16 20.0) (to_q16 4.0)  -- I = 20 / 4 = 5

#eval energyBind {
  agentId := 1,
  charge := to_q16 10.0,
  voltage := to_q16 5.0,
  current := to_q16 2.0,
  power := to_q16 10.0,
  energy := to_q16 50.0,
  time := to_q16 5.0
} {
  agentId := 1,
  workloadDelta := to_q16 5.0,
  resourceLevel := to_q16 6.0,
  duration := to_q16 2.0
}

#eval energyEfficiency (to_q16 40.0) (to_q16 50.0)  -- η = 40/50 = 0.8

#eval powerEfficiency (to_q16 8.0) (to_q16 10.0)  -- η = 8/10 = 0.8

#eval energyPerTask (to_q16 100.0) (to_q16 20.0)  -- E_task = 100/20 = 5

end Semantics.JouleEnergy