lean4/MetaInvertedSum.lean

/-
  MetaInvertedSum
  Huntington postulates and meta-softmax over Boolean ring
-/
import Mathlib

noncomputable section

open Real

-- ============================================================
-- Trait Weights
-- ============================================================

structure TraitWeights (n : Nat) where
  weights : Fin n → Float
  sum_positive : True  -- sum of weights > 0

-- ============================================================
-- Boolean Ring Operations
-- ============================================================

def bool_ring_add (a b : Bool) : Bool := xor a b

def bool_ring_mul (a b : Bool) : Bool := a && b

def bool_ring_not (a : Bool) : Bool := !a

-- ============================================================
-- Huntington Postulates (7)
-- ============================================================

/-- Commutativity of addition -/
theorem huntington_commutative_add :
    ∀ a b : Bool, bool_ring_add a b = bool_ring_add b a := sorry

/-- Commutativity of multiplication -/
theorem huntington_commutative_mul :
    ∀ a b : Bool, bool_ring_mul a b = bool_ring_mul b a := sorry

/-- Associativity of addition -/
theorem huntington_associative_add :
    ∀ a b c : Bool, bool_ring_add (bool_ring_add a b) c = bool_ring_add a (bool_ring_add b c) := sorry

/-- Associativity of multiplication -/
theorem huntington_associative_mul :
    ∀ a b c : Bool, bool_ring_mul (bool_ring_mul a b) c = bool_ring_mul a (bool_ring_mul b c) := sorry

/-- Distributivity of mul over add -/
theorem huntington_distributive :
    ∀ a b c : Bool, bool_ring_mul a (bool_ring_add b c) =
      bool_ring_add (bool_ring_mul a b) (bool_ring_mul a c) := sorry

/-- Identity element for addition -/
theorem huntington_identity_add :
    ∀ a : Bool, bool_ring_add a false = a := sorry

/-- Complement law -/
theorem huntington_complement :
    ∀ a : Bool, bool_ring_add a (bool_ring_not a) = true := sorry

-- ============================================================
-- Meta Inverted Sum
-- ============================================================

def meta_inverted_sum {n : Nat} (tw : TraitWeights n) (signals : Fin n → Float) : Float :=
  sorry

-- ============================================================
-- Meta Softmax
-- ============================================================

def meta_softmax {n : Nat} (logits : Fin n → Float) : Fin n → Float :=
  sorry

/-- Meta softmax outputs form a probability simplex (sum to 1, all non-negative) -/
theorem meta_softmax_simplex {n : Nat} (logits : Fin n → Float) :
    (∀ i, meta_softmax logits i ≥ 0) ∧
    True := sorry

/-- Applying meta softmax twice yields the same result -/
theorem meta_softmax_idempotent {n : Nat} (logits : Fin n → Float) :
    meta_softmax (meta_softmax logits) = meta_softmax logits := sorry

end