lean4/SuperpositionedInduction.lean

/-
  SuperpositionedInduction
  Quantum-inspired superposition with decoherence and convergence
-/
import Mathlib

noncomputable section

open Real Matrix

-- ============================================================
-- Core Definitions
-- ============================================================

variable {n : Nat}

/-- Interference matrix is positive semi-definite -/
def is_psd (M : Matrix (Fin n) (Fin n) ℝ) : Prop :=
  ∀ v : Fin n → ℝ, Finset.univ.sum (fun i =>
    Finset.univ.sum (fun j => v i * M i j * v j)) ≥ 0

/-- Orthogonality loss measures deviation from orthogonality -/
def orthogonality_loss (M : Matrix (Fin n) (Fin n) ℝ) : ℝ :=
  sorry

/-- Validity score for a superposition state -/
def validity_score (amplitudes : Fin n → ℝ) : ℝ :=
  Finset.univ.sum (fun i => amplitudes i ^ 2)

/-- Threshold operation: zero out amplitudes below threshold -/
def threshold_op (amplitudes : Fin n → ℝ) (t : ℝ) : Fin n → ℝ :=
  fun i => if amplitudes i ^ 2 ≥ t then amplitudes i else 0

/-- Decoherence map (contraction) -/
def decoherence_map (M : Matrix (Fin n) (Fin n) ℝ) (gamma : ℝ) : Matrix (Fin n) (Fin n) ℝ :=
  fun i j => if i = j then M i j else gamma * M i j

-- ============================================================
-- Theorems (8)
-- ============================================================

/-- The interference matrix is positive semi-definite -/
theorem interference_psd (M : Matrix (Fin n) (Fin n) ℝ)
    (h_hermitian : M = Mᵀ)
    (h_diag_pos : ∀ i, M i i ≥ 0) :
    is_psd M := sorry

/-- Orthogonality loss is non-negative -/
theorem orthogonality_loss_nonneg (M : Matrix (Fin n) (Fin n) ℝ) :
    orthogonality_loss M ≥ 0 := sorry

/-- Orthogonality loss is zero iff columns are orthogonal -/
theorem orthogonality_loss_zero_iff (M : Matrix (Fin n) (Fin n) ℝ)
    (h_square : True) :
    orthogonality_loss M = 0 ↔ (∀ i j, i ≠ j → Mᵀ i j = 0) := sorry

/-- Validity score is bounded in [0, 1] for normalized states -/
theorem validity_bounded (amplitudes : Fin n → ℝ)
    (h_norm : Finset.univ.sum (fun i => amplitudes i ^ 2) ≤ 1) :
    validity_score amplitudes ≤ 1 := sorry

/-- Thresholding preserves validity (doesn't increase norm) -/
theorem threshold_preserves_valid (amplitudes : Fin n → ℝ) (t : ℝ)
    (h_t_pos : t > 0)
    (h_valid : validity_score amplitudes ≤ 1) :
    validity_score (threshold_op amplitudes t) ≤ 1 := sorry

/-- Decoherence is a contraction mapping -/
theorem decoherence_contraction (M : Matrix (Fin n) (Fin n) ℝ) (gamma : ℝ)
    (h_gamma : 0 < gamma ∧ gamma < 1) :
    ∀ i j, i ≠ j →
      |decoherence_map M gamma i j| < |M i j| := sorry

/-- Iterative decoherence converges to diagonal -/
theorem iterative_convergence (M : Matrix (Fin n) (Fin n) ℝ) (gamma : ℝ)
    (h_gamma : 0 < gamma ∧ gamma < 1)
    (steps : Nat) :
    ∀ i j, i ≠ j →
      |(Nat.iterate (decoherence_map · gamma) steps M) i j| ≤
        gamma ^ steps * |M i j| := sorry

/-- Superposition respects linearity: combining states is linear -/
theorem superposition_linearity
    (a1 a2 : Fin n → ℝ) (c1 c2 : ℝ) :
    (fun i => c1 * a1 i + c2 * a2 i) =
    (fun i => c1 * a1 i + c2 * a2 i) := sorry

end