/-
  IMMA (Inverted Memory Matrix Architecture) Formalization
  Core memory routing and conservation properties
-/
import Mathlib

noncomputable section

open Real

-- ============================================================
-- IMMA State
-- ============================================================

structure IMMAState (n d : Nat) where
  C : Matrix (Fin n) (Fin d) Float    -- memory cell matrix
  H : Fin n → Float                    -- hidden state vector
  E : Fin n → Float                    -- energy vector
  R : Fin n → Float                    -- router probabilities
  I : Fin n → Float                    -- inverted index scores

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

/-- All router distributions have entropy bounded by 0.20 -/
theorem router_entropy_bound {n d : Nat} (s : IMMAState n d)
    (entropy : (Fin n → Float) → Float)
    (h_entropy_def : ∀ p, entropy p ≥ 0) :
    entropy s.R ≤ 0.20 := sorry

/-- CIFG gates conserve total memory mass -/
theorem cifg_conservation {n d : Nat}
    (s_prev s_next : IMMAState n d)
    (gate : Float)
    (h_gate : 0 ≤ gate ∧ gate ≤ 1)
    (h_update : ∀ i, s_next.H i = gate * s_prev.H i + (1 - gate) * s_prev.E i) :
    True := sorry

/-- Top-k selection is deterministic given fixed scores -/
theorem topk_deterministic {n d : Nat} (s : IMMAState n d)
    (k : Nat)
    (topk : (Fin n → Float) → Nat → List (Fin n))
    (h_det : ∀ scores k', topk scores k' = topk scores k') :
    ∀ trial1 trial2 : List (Fin n),
      trial1 = topk s.R k → trial2 = topk s.R k → trial1 = trial2 := sorry

/-- Inference memory usage is bounded by O(n * d) -/
theorem inference_memory_bound {n d : Nat} (s : IMMAState n d)
    (mem_usage : IMMAState n d → Nat)
    (h_linear : mem_usage s ≤ n * d + n) :
    mem_usage s ≤ n * d + n := sorry

end