/-
  BURT (Bi-encoder Unified Retrieval Transformer) Formalization
  Core retrieval and ranking properties
-/
import Mathlib

noncomputable section

open Real

-- ============================================================
-- BURT State
-- ============================================================

structure BURTState (n d : Nat) where
  Q : Matrix (Fin n) (Fin d) Float    -- query embeddings
  D : Matrix (Fin n) (Fin d) Float    -- document embeddings
  I : Fin n → Float                    -- inverted index scores
  R : Fin n → Fin n                    -- ranking permutation
  C : Float                            -- CIFG gate value
  E : Float                            -- energy

-- ============================================================
-- Core Theorems
-- ============================================================

/-- Router entropy is bounded by the entropy ceiling -/
theorem router_entropy_bound {n d : Nat} (s : BURTState n d)
    (entropy : (Fin n → Float) → Float)
    (router_dist : Fin n → Float)
    (h_from_state : ∀ i, router_dist i = s.I i) :
    entropy router_dist ≤ 0.20 := sorry

/-- CIFG gate conserves memory during retrieval updates -/
theorem cifg_memory_conservation {n d : Nat}
    (s : BURTState n d)
    (h_gate : 0 ≤ s.C ∧ s.C ≤ 1)
    (mem_prev mem_next : Float)
    (h_update : mem_next = s.C * mem_prev + (1 - s.C) * s.E) :
    mem_next ≤ max mem_prev s.E := sorry

/-- The ranking function produces a valid permutation -/
theorem ranking_is_permutation {n d : Nat} (s : BURTState n d)
    (h_inj : Function.Injective s.R) :
    Function.Bijective s.R := sorry

/-- Retrieval has sub-quadratic complexity via inverted index -/
theorem retrieval_complexity {n d : Nat} (s : BURTState n d)
    (retrieval_flops : Nat)
    (h_indexed : retrieval_flops ≤ n * Nat.log n + d) :
    retrieval_flops < n * n := sorry

end