import Mathlib.Data.Nat.Basic
import Mathlib.Tactic
import Semantics.FixedPoint

open Semantics

/-! # Domain-Bound Signal Transform: Evolutionary Path Finding

This module formalizes the Long-Term Evolution Experiment (LTEE) as a
domain-bound signal transform where genetic signals are transformed into
phenotypic signals through automatic path finding (natural selection).

**Domain-Bound Signal Transform**:
- Input domain: Genetic signals (mutations, gene changes, genomic variations)
- Output domain: Phenotypic signals (fitness, cell size, metabolic capabilities)
- Transform mechanism: Automatic path finding via natural selection
- Domain boundaries: LTEE experimental constraints (glucose-limited DM25 medium)

**Signal Flow**:
1. Genetic signal enters domain (de novo mutation)
2. Automatic path finding evaluates fitness landscape
3. Selection amplifies beneficial signals, attenuates deleterious
4. Phenotypic signal emerges (fitness change, morphological adaptation)
5. Domain boundary: frozen fossil record preserves signal state

**LTEE Experimental Parameters**:
- 12 populations from same ancestral strain (1988)
- ~6.67 generations per day (100-fold growth)
- Samples frozen every 500 generations (75 days)
- Over 73,000 generations by 2020
- Fitness increase: ~70% faster than ancestor by 20,000 generations

Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: All definitions must have eval witnesses or theorems.
-/

namespace EvolutionaryTransfold

/-- Genetic signal type (input domain).-/
inductive GeneticSignal where
  | point       -- Single nucleotide change
  | insertion   -- Insertion sequence element
  | deletion    -- Deletion
  | duplication -- Gene duplication
  deriving Repr, DecidableEq, Inhabited

/-- Genetic signal state (input domain).
  Represents the input signal to the domain-bound transform.
-/
structure GeneticSignalState where
  signalAmplitude : Nat     -- Number of mutations (signal strength)
  signalType : GeneticSignal -- Type of genetic signal
  mutatorAmplification : Bool -- Whether mutator amplifies signal
  citCapability : Bool      -- Cit+ metabolic signal capability
  deriving Repr, Inhabited

/-- Phenotypic signal state (output domain).
  Represents the output signal from the domain-bound transform.
-/
structure PhenotypicSignalState where
  fitnessSignal : Q16_16    -- Fitness output signal amplitude
  sizeSignal : Q16_16       -- Cell size output signal
  densitySignal : Q16_16    -- Population density output signal
  deriving Repr, Inhabited

/-- Domain boundary constraints (LTEE experimental limits).-/
structure DomainBoundary where
  maxPopulationSize : Nat   -- Maximum cells in 10mL culture (500M)
  glucoseConcentration : Nat -- Glucose limit (25 mg/L)
  citrateConcentration : Nat -- Citrate abundance (~275 mg/L)
  temperature : Nat         -- Incubator temperature (37°C)
  deriving Repr, Inhabited

/-- Time parameter for signal transform.-/
structure SignalTime where
  elapsedGenerations : Nat   -- Total generations elapsed
  sampleFrozen : Bool        -- Whether signal state is frozen (boundary condition)
  deriving Repr, Inhabited

/-- Domain-bound signal transform: genetic signal → phenotypic signal.

  This is the signal transform performed by evolution:
  1. Genetic signal (discrete mutations) → Phenotypic signal (continuous fitness)
  2. Automatic path finding via natural selection amplifies/attenuates signals
  3. Domain boundary constraints limit signal propagation

  Based on LTEE data showing power-law fitness increase over 60,000+ generations.
-/
def evolutionarySignalTransform (genetic : GeneticSignalState) (time : SignalTime) : PhenotypicSignalState :=
  let fitnessBase := Q16_16.ofInt 100  -- Ancestor baseline
  let fitnessIncrease := Q16_16.mul (Q16_16.ofInt genetic.signalAmplitude) (Q16_16.ofInt 2)
  let fitnessSignal := Q16_16.add fitnessBase fitnessIncrease
  let sizeIncrease := if genetic.mutatorAmplification then Q16_16.ofInt 20 else Q16_16.ofInt 10
  let sizeSignal := Q16_16.add (Q16_16.ofInt 100) sizeIncrease
  let densityDecrease := Q16_16.div (Q16_16.ofInt 500) (Q16_16.add (Q16_16.ofInt 1) (Q16_16.ofInt (time.elapsedGenerations / 10000)))
  { fitnessSignal, sizeSignal, densitySignal := densityDecrease }

/-- Theorem: Signal transform preserves signal amplitude invariants.

  If two genetic signals have same amplitude,
  their phenotypic signals have same fitness baseline.
  This reflects LTEE finding that clonal markers persist by descent.
-/
theorem signalAmplitudePreserved (genetic1 genetic2 : GeneticSignalState) (time : SignalTime) :
  genetic1.signalAmplitude = genetic2.signalAmplitude →
  let phen1 := evolutionarySignalTransform genetic1 time
  let phen2 := evolutionarySignalTransform genetic2 time
  phen1.fitnessSignal = phen2.fitnessSignal := by
  intro h
  simp [evolutionarySignalTransform, h]

/-- LTEE signal rate: ~6.67 generations per day.

  Based on LTEE experimental protocol: 100-fold growth = ~6.67 doublings.
-/
def lteeSignalRatePerDay : Nat := 20 / 3  -- 6.67 generations/day

/-- LTEE sampling interval: every 500 generations (75 days).
  This creates periodic boundary conditions for signal state preservation.
-/
def lteeSamplingInterval : Nat := 500

/-- Theorem: Sampling preserves signal state modulo.

  Frozen samples are taken every 500 generations, creating a periodic
  boundary condition that allows signal state reconstruction across time.
-/
theorem samplingPreservesSignalModulo (generations : Nat) :
  generations % lteeSamplingInterval = 0 →
  let sample : SignalTime := { elapsedGenerations := generations, sampleFrozen := true }
  sample.elapsedGenerations % lteeSamplingInterval = 0 := by
  intro h
  exact h

/-- Domain boundary verification: signal within LTEE constraints.

  Verifies that phenotypic signal stays within domain boundaries.
-/
def withinDomainBoundary (phenotype : PhenotypicSignalState) (boundary : DomainBoundary) : Bool :=
  let fitnessOk := phenotype.fitnessSignal.toInt ≤ 200 * Q16_16.scale  -- Raw Q16.16 fitness bound
  let densityOk := phenotype.densitySignal.toInt ≤ boundary.maxPopulationSize
  fitnessOk && densityOk

/-- The checked boundary gate for the transformed signal.

  The current transform does not prove a universal biological bound from the
  raw inputs alone.  Promotion therefore goes through this explicit Boolean
  gate instead of an unbounded theorem claim.
-/
def domainBoundaryGate (genetic : GeneticSignalState) (time : SignalTime) (boundary : DomainBoundary) : Bool :=
  withinDomainBoundary (evolutionarySignalTransform genetic time) boundary

/-- Theorem: accepted domain-boundary states are exactly the checked gate.

  This preserves the receipt boundary without guessing an unproven global
  biological constraint.
-/
theorem domainBoundaryGateSound (genetic : GeneticSignalState) (time : SignalTime) (boundary : DomainBoundary) :
  domainBoundaryGate genetic time boundary =
    withinDomainBoundary (evolutionarySignalTransform genetic time) boundary := by
  rfl

/-- Power-law signal amplification model (LTEE finding).

  LTEE data shows fitness signal amplification follows power law: signal ∝ t^α
  where α < 1, indicating ever-slowing but unbounded signal growth.
-/
def powerLawSignalAmplification (generations : Nat) : Q16_16 :=
  let genFloat := Q16_16.ofInt generations
  Q16_16.mul genFloat (Q16_16.sqrt genFloat)  -- Simplified power law t^1.5

/-- Executable monotonicity gate for the current fixed-point approximation.

  `Q16_16.sqrt` is an implementation-level approximation through Float at the
  boundary, so this module records monotonicity as a checked witness rather
  than an unconditional theorem.
-/
def powerLawMonotonicGate (g1 g2 : Nat) : Bool :=
  if g1 < g2 then
    powerLawSignalAmplification g1 < powerLawSignalAmplification g2
  else
    true

/-- The monotonicity gate is definitionally true for non-increasing inputs. -/
theorem powerLawMonotonicGateInactive (g1 g2 : Nat) (h : ¬ g1 < g2) :
  powerLawMonotonicGate g1 g2 = true := by
  simp [powerLawMonotonicGate, h]

/-- The monotonicity gate exposes the exact checked fixed-point comparison. -/
theorem powerLawMonotonicGateActive (g1 g2 : Nat) (h : g1 < g2) :
  powerLawMonotonicGate g1 g2 =
    decide (powerLawSignalAmplification g1 < powerLawSignalAmplification g2) := by
  simp [powerLawMonotonicGate, h]

/-- The complete Domain-Bound Signal Transform as a single expression.

  T(genetic_signal, time) = phenotypic_signal
  where automatic path finding performs signal transformation:
  1. Genetic signal (discrete mutations) → Phenotypic signal (continuous fitness)
  2. Automatic path finding via natural selection amplifies/attenuates
  3. Domain boundary constraints limit signal propagation
  4. Frozen boundary conditions enable signal state reconstruction

  The invariant root is: **signal amplitude under selection-driven automatic path finding**.
-/
def DomainBoundSignalTransform (genetic : GeneticSignalState) (time : SignalTime) : PhenotypicSignalState :=
  evolutionarySignalTransform genetic time

def sampleGeneticSignal : GeneticSignalState :=
  { signalAmplitude := 10
  , signalType := GeneticSignal.point
  , mutatorAmplification := false
  , citCapability := false
  }

def sampleSignalTime : SignalTime :=
  { elapsedGenerations := 500
  , sampleFrozen := true
  }

def sampleBoundary : DomainBoundary :=
  { maxPopulationSize := 500 * Q16_16.scale
  , glucoseConcentration := 25
  , citrateConcentration := 275
  , temperature := 37
  }

#eval domainBoundaryGate sampleGeneticSignal sampleSignalTime sampleBoundary
-- Expected: true

#eval powerLawMonotonicGate 10 20
-- Expected: true for the current fixed-point approximation

end EvolutionaryTransfold
