lean4/HumorEntropy.lean

/-
  Copyright (c) 2026 BEL ESPRIT D ACCORD TRUST HOLDINGS INC
  All rights reserved.
-/
-- HumorEntropy.lean
-- Formal verification of HUMOR-ENTROPY-ALGORITHM proof obligations

namespace HumorEntropy

open Real
open List
open Finset

/-- Primitive Definitions -/

-- Embedding vector as finite sequence of reals
def Embedding (D : ℕ) := Fin D → ℝ

-- Projection to first 1024 dimensions
def Project1024 {D : ℕ} (e : Embedding D) : Embedding 1024 :=
  fun i => e ⟨i.val, by
    have : i.val < D := by
      have h : i.val < 1024 := Fin.is_lt i
      have h₂ : 1024 ≤ D ∨ D < 1024 := by omega
      cases h₂ with
      | inl h₂ => omega
      | inr h₂ =>
        exfalso
        have : i.val < D := by
          have h₃ : i.val < 1024 := Fin.is_lt i
          omega
        omega
    exact this⟩

-- L2 norm
def L2Norm {D : ℕ} (v : Embedding D) : ℝ :=
  Real.sqrt (∑ i : Fin D, (v i) ^ 2)

-- L2 normalization to unit sphere
def NormalizeL2 {D : ℕ} (v : Embedding D) : Embedding D :=
  fun i => if h : L2Norm v > 0 then v i / L2Norm v else 0

-- Cosine distance on unit sphere
def CosineDistance {D : ℕ} (x y : Embedding D) : ℝ :=
  1 - ∑ i : Fin D, x i * y i

-- Shannon entropy in nats
def ShannonEntropy (probs : List ℝ) : ℝ :=
  probs.foldr (fun p acc => if p > 0 then acc + -p * Real.log p else acc) 0

-- Sigmoid function
def Sigmoid (x : ℝ) : ℝ :=
  1 / (1 + Real.exp (-x))

/- NAND Boolean Kernel -/

def NAND (a b : Bool) : Bool := !(a && b)

def NotNAND (x : Bool) : Bool := NAND x x

def AndNAND (a b : Bool) : Bool := NAND (NAND a b) (NAND a b)

def OrNAND (a b : Bool) : Bool := NAND (NAND a a) (NAND b b)

/- Proof Obligations as Theorems -/

-- P1: Normalization preserves unit norm
theorem p1_normalization {D : ℕ} (v : Embedding D) :
  L2Norm (NormalizeL2 v) = 1 ∨ L2Norm v = 0 := by sorry

-- P2: Entropy bound implies entropy_ok
theorem p2_entropy_bound (H_local : ℝ) (h : H_local ≤ 0.20) :
  NotNAND (H_local > (0.20 : ℝ)) = true := by sorry

-- P3: Humor potential monotonic in incongruity (fixed entropy)
theorem p3_monotonicity_incongruity
  (δ₁ δ₂ H : ℝ) (w_dot : ℝ)
  (hδ : δ₁ < δ₂) (hH : H = H) :
  δ₁ * (1 - H / 0.20) * Sigmoid w_dot < δ₂ * (1 - H / 0.20) * Sigmoid w_dot := by sorry

-- P4: Humor potential decreasing in entropy (fixed incongruity)
theorem p4_entropy_penalty
  (δ H₁ H₂ : ℝ) (w_dot : ℝ)
  (hH : H₁ < H₂) (hδ : δ = δ) :
  δ * (1 - H₁ / 0.20) * Sigmoid w_dot > δ * (1 - H₂ / 0.20) * Sigmoid w_dot := by sorry

-- P5: DAG compliance - output depends only on pipeline stages
structure PipelineState where
  input : Embedding 1024
  memory : List (Embedding 1024)
  retrieval : ℝ -- H_local
  transform : ℝ -- humor_potential
  constraint : Bool -- benign
  proof : ProofCertificate
  output : Output

structure ProofCertificate where
  h_local : ℝ
  delta : ℝ
  humor_potential : ℝ
  benign : Bool
  entropy_ok : Bool
  incongruity_ok : Bool

structure Output where
  is_humorous : Bool
  score : ℝ

-- P7: Determinism - same inputs produce same outputs
theorem p7_determinism
  (e₁ e₂ : Embedding 1024) (c₁ c₂ : Embedding 1024) (w₁ w₂ : Embedding 1024)
  (n₁ n₂ : List (Embedding 1024))
  (he : e₁ = e₂) (hc : c₁ = c₂) (hw : w₁ = w₂) (hn : n₁ = n₂) :
  humor_entropy_instruct e₁ c₁ w₁ n₁ = humor_entropy_instruct e₂ c₂ w₂ n₂ := by sorry

-- P8: Sovereign constraint - active implies trusted
structure Agent where
  id : String
  role : GlyphUnit
  entropy : ℝ
  trusted : Bool
  active : Bool

inductive GlyphUnit
  | Cognition | Knowledge | Search | Constraint
  | Transformation | Memory | Proof | Interface

theorem p8_sovereign (a : Agent) :
  a.active → a.trusted := by sorry

-- Entropy bound invariant
theorem entropy_invariant (a : Agent) :
  a.entropy ≤ 0.20 := by sorry

/- Main Algorithm Specification -/

def humor_entropy_instruct
  (embedding : Embedding 1024)
  (context : Embedding 1024)
  (weights : Embedding 1024)
  (neighborhood : List (Embedding 1024))
  : Output := by
  let x := embedding
  let c := context
  let w := weights

  -- Memory + Retrieval: Local entropy
  let probs := neighborhood.map (fun y => Real.exp (-(CosineDistance x y)))
  let sum_probs := probs.foldl (fun acc p => acc + p) 0
  let normalized_probs := probs.map (fun p => p / sum_probs)
  let h_local := ShannonEntropy normalized_probs

  -- Transform
  let delta := CosineDistance c x
  let dot_weight := ∑ i : Fin 1024, x i * w i
  let humor_potential := delta * (1 - h_local / 0.20) * Sigmoid dot_weight

  -- Constraint: NAND-only logic
  let entropy_ok : Bool := NotNAND (h_local > (0.20 : ℝ))
  let incongruity_high : Bool := (delta : ℝ) > 0.15
  let incongruity_low : Bool := (delta : ℝ) < 0.65
  let incongruity_ok : Bool := AndNAND incongruity_high incongruity_low
  let benign : Bool := AndNAND entropy_ok incongruity_ok

  -- Proof
  let _certificate : ProofCertificate := ⟨h_local, delta, humor_potential, benign, entropy_ok, incongruity_ok⟩

  -- Output
  exact ⟨benign, if benign then humor_potential else 0⟩

/- Certificate Consistency Theorem -/

theorem certificate_consistency
  (embedding context weights : Embedding 1024)
  (neighborhood : List (Embedding 1024)) :
  let result := humor_entropy_instruct embedding context weights neighborhood
  result.is_humorous = true ∨ result.score = 0 := by sorry

/- Adversarial Properties -/

-- T3: Identical context yields zero incongruity
theorem t3_identical_context :
  ∀ (x : Embedding 1024),
  CosineDistance x x = 0 := by sorry

-- T4: Orthogonal context yields maximum incongruity
theorem t4_orthogonal_context :
  ∃ (x y : Embedding 1024), CosineDistance x y = 1 := by sorry

-- T9: Determinism (computational)
theorem t9_determinism_computational :
  ∀ (e c w : Embedding 1024) (n : List (Embedding 1024)),
  humor_entropy_instruct e c w n = humor_entropy_instruct e c w n := by rfl

end HumorEntropy