/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

MathGPT.lean — Mathematical Rigor Enforcement System

This module provides automated mathematical verification to prevent
incorrect formulations from entering the codebase.

Core principle: All equations must be physically consistent before implementation.
-/

import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Real.Basic

namespace MathGPT

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Physical Law Registry — Immutable Truth Sources
-- ═══════════════════════════════════════════════════════════════════════════

/-- A physical law that must be respected by all equations -/
structure PhysicalLaw where
  name : String
  statement : String
  mathematicalForm : String
  domain : String  -- "thermodynamics", "information_theory", "quantum", etc.
  violations : List String  -- Common incorrect formulations
  deriving Repr

/-- LANDAUER'S PRINCIPLE — Core constraint on information erasure
    
    The minimum energy to erase one bit of information at temperature T.
    This is the foundation of all information-thermodynamics in OTOM.
    
    Mathematical form: E_min = k_B · T · ln(N)
    
    Common violations to reject:
    1. E ∝ 1/ln(N) — "inverse cost" (wrong!)
    2. E ∝ ln(1/N) — negative entropy (wrong!)
    3. E independent of N — constant cost (wrong!)
    -/
def landauerPrinciple : PhysicalLaw := {
  name := "Landauer's Principle",
  statement := "Minimum energy to erase information scales as ln(N)",
  mathematicalForm := "E_min = k_B · T · ln(N)",
  domain := "thermodynamics",
  violations := [
    "E ∝ 1/ln(N) — inverse cost violates physics",
    "E ∝ -ln(N) — negative entropy impossible",
    "E constant — ignores alphabet size"
  ]
}

/-- SHANNON ENTROPY — Information content bound
    
    Mathematical form: H = -Σ p · log₂(p)
    Maximum: log₂(N) for uniform distribution
    -/
def shannonEntropy : PhysicalLaw := {
  name := "Shannon Entropy",
  statement := "Information content bounded by log(N)",
  mathematicalForm := "H = -Σ p · log₂(p) ≤ log₂(N)",
  domain := "information_theory",
  violations := [
    "H > log₂(N) — exceeds maximum",
    "H < 0 — negative entropy impossible"
  ]
}

/-- CONSERVATION OF ENERGY — First law of thermodynamics
    
    Energy cannot be created or destroyed, only converted.
    -/
def conservationOfEnergy : PhysicalLaw := {
  name := "Conservation of Energy",
  statement := "Total energy constant in isolated system",
  mathematicalForm := "ΔE_total = 0",
  domain := "thermodynamics",
  violations := [
    "ΔE < 0 — energy destruction",
    "ΔE > 0 — energy creation"
  ]
}

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Equation Validator — Automated Rigor Checking
-- ═══════════════════════════════════════════════════════════════════════════

/-- Validation result for an equation -/
inductive ValidationResult
  | valid (reason : String)
  | invalid (reason : String) (lawViolated : String)
  | warning (message : String)
  deriving Repr

/-- Check if equation respects Landauer scaling
    
    E(N) must be:
    1. Monotonically increasing in N
    2. Proportional to ln(N), not 1/ln(N)
    3. Non-negative for all N ≥ 2
    -/
def checkLandauerScaling (costFunction : ℕ → ℝ) : ValidationResult :=
  -- Check monotonicity: N₁ < N₂ → cost(N₁) < cost(N₂)
  let monoCheck := ∀ N₁ N₂ : ℕ, N₁ ≥ 2 → N₂ ≥ 2 → N₁ < N₂ → costFunction N₁ < costFunction N₂
  
  -- Check proportionality: cost(N) ∝ ln(N)
  let propCheck := ∃ k : ℝ, k > 0 ∧ ∀ N : ℕ, N ≥ 2 → costFunction N = k * Real.log N
  
  -- Check non-negativity
  let nonNegCheck := ∀ N : ℕ, N ≥ 2 → costFunction N ≥ 0
  
  if ¬monoCheck then
    ValidationResult.invalid 
      "Cost decreases with alphabet size — violates Landauer monotonicity"
      "Landauer's Principle"
  else if ¬nonNegCheck then
    ValidationResult.invalid
      "Negative thermodynamic cost — physically impossible"
      "Landauer's Principle"
  else
    ValidationResult.valid "Landauer scaling respected"

/-- Check for common incorrect formulations -/
def checkCommonErrors (equation : String) : List ValidationResult :=
  let errors := [
    ("/lnN", "Inverse logarithmic scaling — violates Landauer"),
    ("1/ln", "Reciprocal cost — physically absurd"),
    ("ln(1/N)", "Negative entropy argument — impossible"),
    ("-lnN", "Negative cost — violates second law"),
    ("/ln N", "Spaced inverse form — still wrong"),
    ("denominator.*ln", "ln in denominator — check Landauer consistency")
  ]
  
  errors.filterMap (λ (pattern, reason) =>
    if equation.contains pattern then
      some (ValidationResult.invalid reason "Landauer's Principle")
    else
      none
  )

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Universal Field Validator — Specific to EQUATION #0
-- ═══════════════════════════════════════════════════════════════════════════

/-- Validate Universal Field Φ against physical laws
    
    Two forms must be checked:
    1. Cost form: Φ = Σ w·lnN - Σ v·lnN  ← MUST pass Landauer
    2. Efficiency form: Φ = Σ w·h/lnN - Σ v·p/lnN  ← Inverse is correct here
    -/
def validateUniversalField (equation : String) : ValidationResult :=
  -- Check for old (wrong) cost form
  if equation.contains "w/lnN" ∨ equation.contains "wᵢ/lnNᵢ" then
    ValidationResult.invalid
      "CRITICAL: lnN in denominator for cost form violates Landauer. " ++
      "Cost must be w·lnN, not w/lnN. " ++
      "Higher alphabet = higher energy cost."
      "Landauer's Principle"
  
  -- Check for correct cost form
  else if equation.contains "w·lnN" ∨ equation.contains "w*lnN" ∨ equation.contains "w * ln" then
    ValidationResult.valid
      "Correct Landauer scaling: cost ∝ lnN. " ++
      "Higher alphabet = higher thermodynamic cost."
  
  -- Check for efficiency form (lnN in denominator is correct here)
  else if equation.contains "h/lnN" ∨ equation.contains "hᵢ/lnNᵢ" then
    ValidationResult.valid
      "Efficiency form correct: h/lnN = quality per unit cost. " ++
      "Efficiency decreases as cost increases."
  
  -- Ambiguous or unrecognizable
  else
    ValidationResult.warning
      "Equation form unclear — manual review required"

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Pre-Commit Hook — Block Bad Math Before Entry
-- ═══════════════════════════════════════════════════════════════════════════

/-- Pre-commit validation for any new equation
    
    This function should be called before any equation is:
    - Added to math_entities.db
    - Committed to git
    - Added to MATH_MODEL_MAP
    - Implemented in Lean
    - Published in papers
    
    Returns: (is_valid, reasons)
    -/
def preCommitValidation (equation : String) (author : String) : (Bool × List String) :=
  let results := checkCommonErrors equation
  let universalCheck := [validateUniversalField equation]
  
  let allResults := results ++ universalCheck
  
  let errors := allResults.filterMap (λ r =>
    match r with
    | ValidationResult.invalid reason law => some s!"[VIOLATION: {law}] {reason}"
    | _ => none
  )
  
  let warnings := allResults.filterMap (λ r =>
    match r with
    | ValidationResult.warning msg => some s!"[WARNING] {msg}"
    | _ => none
  )
  
  let isValid := errors.isEmpty
  
  let report := if isValid then
    [s!"✅ VALIDATED by MathGPT for {author}",
     s!"Equation passes all physical law checks"]
    ++ warnings
  else
    [s!"❌ REJECTED by MathGPT for {author}",
     s!"Equation violates physical laws — cannot be committed"]
    ++ errors
    ++ warnings
    ++ ["",
        "FIX REQUIRED: Ensure equation respects:",
        s!"  - {landauerPrinciple.name}: {landauerPrinciple.mathematicalForm}",
        "",
        "Common fixes:",
        "  - Change w/lnN to w·lnN (cost form)",
        "  - Or explicitly use h/lnN for efficiency (inverse is correct there)"]
  
  (isValid, report)

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Automated Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

-- Example: Correct cost form
def exampleCorrectCost : String := "Φ = Σ w·lnN - Σ v·lnN"
#eval preCommitValidation exampleCorrectCost "Builder"

-- Example: Incorrect (old) cost form — SHOULD BE REJECTED
def exampleIncorrectCost : String := "Φ = Σ w/lnN + Σ v/lnN"
#eval preCommitValidation exampleIncorrectCost "Builder"

-- Example: Correct efficiency form
def exampleCorrectEfficiency : String := "Φ = Σ w·h/lnN - Σ v·p/lnN"
#eval preCommitValidation exampleCorrectEfficiency "Builder"

-- Example: Ambiguous form — SHOULD WARN
def exampleAmbiguous : String := "Φ = Σ wN"
#eval preCommitValidation exampleAmbiguous "Builder"

end MathGPT