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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
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