(*
 * Entropy Distribution Validation — Formal Specification
 * Ahmad Ali Parr · 2026-08-03
 *
 * Formal verification of quantum entropy source validation via ±10% NISQ tolerance.
 *
 * ## Specification
 *
 * Given byte stream from ANU QRNG (real quantum vacuum fluctuations):
 * 1. Count total bits
 * 2. Count ones
 * 3. Calculate ones_ratio = ones / total_bits
 * 4. Verify |ones_ratio - 0.5| ≤ TOLERANCE (default 0.10)
 *
 * ## Properties to Prove
 *
 * 1. **Soundness**: If validation passes, source is statistically random (NISQ grade)
 * 2. **Completeness**: True random source passes with high probability
 * 3. **Rejection**: Non-random sources (all 0s, all 1s, patterns) fail
 * 4. **Tolerance Bound**: ±10% catches bias while allowing quantum noise
 * 5. **Monotonicity**: Stricter tolerance → fewer false positives
 *
 * ## Reference Implementation
 *
 * JavaScript (src/quantum_entropy.mjs):
 * ```javascript
 * function validateDistribution (uint16s) {
 *   const bytes       = []
 *   for (const v of uint16s) { bytes.push((v >> 8) & 0xff, v & 0xff) }
 *   const totalBits   = bytes.length * 8
 *   let ones          = 0
 *   for (const b of bytes) {
 *     let x = b
 *     while (x) { ones += x & 1; x >>= 1 }
 *   }
 *   const onesRatio = ones / totalBits
 *   const passed    = Math.abs(onesRatio - 0.5) <= TOLERANCE
 *   return { totalBits, ones, zeros: totalBits - ones, onesRatio, passed }
 * }
 * ```
 *)

Require Import Coq.Reals.Reals.
Require Import Coq.Lists.List.
Require Import Coq.Arith.Arith.
Require Import Coq.QArith.QArith.
Require Import Coq.QArith.Qabs.
Import ListNotations.
Open Scope R_scope.

(* ═══════════════════════════════════════════════════════════════════════════
   CORE TYPES
   ═══════════════════════════════════════════════════════════════════════════ *)

(* Byte: 8-bit value *)
Definition Byte := { n : nat | n < 256 }.

(* Bit: 0 or 1 *)
Inductive Bit := Zero | One.

(* Entropy source type *)
Inductive EntropySource :=
  | QuantumVacuum   : EntropySource  (* ANU QRNG - true quantum *)
  | CSPRNG          : EntropySource  (* Cryptographic fallback *)
  | Deterministic   : EntropySource. (* Non-random (all 0s, patterns) *)

(* Validation result *)
Record ValidationResult := {
  total_bits : nat;
  ones_count : nat;
  zeros_count : nat;
  ones_ratio : R;
  passed : bool
}.

(* Tolerance constant (±10% NISQ grade) *)
Definition TOLERANCE : R := 0.10.

(* Expected ratio for true random source *)
Definition EXPECTED_RATIO : R := 0.5.

(* ═══════════════════════════════════════════════════════════════════════════
   BIT EXTRACTION
   ═══════════════════════════════════════════════════════════════════════════ *)

(* Extract bits from byte (LSB first) *)
Fixpoint byte_to_bits (b : nat) (fuel : nat) : list Bit :=
  match fuel with
  | O => []
  | S fuel' =>
      let bit := if Nat.even b then Zero else One in
      bit :: byte_to_bits (Nat.div b 2) fuel'
  end.

Definition byte_to_8bits (b : nat) : list Bit :=
  byte_to_bits b 8.

(* Count ones in bit list *)
Fixpoint count_ones (bits : list Bit) : nat :=
  match bits with
  | [] => 0
  | Zero :: rest => count_ones rest
  | One :: rest => S (count_ones rest)
  end.

(* ═══════════════════════════════════════════════════════════════════════════
   VALIDATION ALGORITHM
   ═══════════════════════════════════════════════════════════════════════════ *)

(* Extract all bits from byte list *)
Definition bytes_to_bits (bytes : list nat) : list Bit :=
  flat_map byte_to_8bits bytes.

(* Validate entropy distribution *)
Definition validate_distribution (bytes : list nat) (tolerance : R) : ValidationResult :=
  let bits := bytes_to_bits bytes in
  let total := length bits in
  let ones := count_ones bits in
  let zeros := total - ones in
  let ratio := if Nat.eqb total 0 then 0 else INR ones / INR total in
  let deviation := Rabs (ratio - EXPECTED_RATIO) in
  let passed := if Rle_dec deviation tolerance then true else false in
  {|
    total_bits := total;
    ones_count := ones;
    zeros_count := zeros;
    ones_ratio := ratio;
    passed := passed
  |}.

(* ═══════════════════════════════════════════════════════════════════════════
   THEOREMS
   ═══════════════════════════════════════════════════════════════════════════ *)

(* T1: Bit extraction is total (always produces 8 bits per byte) *)
Theorem byte_to_8bits_length : forall b,
  b < 256 ->
  length (byte_to_8bits b) = 8.
Proof.
  intros b Hbound.
  unfold byte_to_8bits.
  (* Induction on fuel *)
  unfold byte_to_bits.
  simpl.
  (* Compute: 8 recursive calls produce 8 bits *)
  repeat (destruct (Nat.even _); simpl); reflexivity.
Qed.

(* T2: Ones + Zeros = Total *)
Theorem ones_plus_zeros_eq_total : forall bytes vr,
  vr = validate_distribution bytes TOLERANCE ->
  ones_count vr + zeros_count vr = total_bits vr.
Proof.
  intros bytes vr Hvr.
  unfold validate_distribution in Hvr.
  subst vr.
  simpl.
  lia.
Qed.

(* Helper: count_ones ≤ length *)
Lemma count_ones_le_length : forall bits,
  count_ones bits <= length bits.
Proof.
  induction bits.
  - simpl. lia.
  - simpl. destruct a; simpl; lia.
Qed.

(* T3: Ratio bounds [0, 1] *)
Theorem ratio_in_unit_interval : forall bytes vr,
  vr = validate_distribution bytes TOLERANCE ->
  total_bits vr > 0 ->
  0 <= ones_ratio vr <= 1.
Proof.
  intros bytes vr Hvr Htotal.
  unfold validate_distribution in Hvr.
  subst vr.
  simpl.
  split.
  - (* 0 <= ratio *)
    apply Rdiv_le_0_compat.
    + apply pos_INR.
    + apply lt_INR. lia.
  - (* ratio <= 1 *)
    apply Rdiv_le_1.
    + apply lt_INR. lia.
    + apply le_INR.
      apply count_ones_le_length.
Qed.

(* T4: All zeros fails validation (unless tolerance ≥ 0.5) *)
Theorem all_zeros_fails : forall n,
  n > 0 ->
  TOLERANCE < 0.5 ->
  passed (validate_distribution (repeat 0 n) TOLERANCE) = false.
Proof.
  intros n Hn Htol.
  unfold validate_distribution.
  simpl.
  (* All zeros → ones_ratio = 0 *)
  (* |0 - 0.5| = 0.5 > TOLERANCE *)
  admit.
Admitted.

(* T5: All ones fails validation (unless tolerance ≥ 0.5) *)
Theorem all_ones_fails : forall n,
  n > 0 ->
  TOLERANCE < 0.5 ->
  passed (validate_distribution (repeat 255 n) TOLERANCE) = false.
Proof.
  intros n Hn Htol.
  unfold validate_distribution.
  simpl.
  (* All ones (0xFF) → ones_ratio = 1.0 *)
  (* |1.0 - 0.5| = 0.5 > TOLERANCE *)
  admit.
Admitted.

(* T6: Stricter tolerance → fewer accepted sources *)
Theorem stricter_tolerance_stronger : forall bytes t1 t2,
  t1 < t2 ->
  passed (validate_distribution bytes t1) = true ->
  passed (validate_distribution bytes t2) = true.
Proof.
  intros bytes t1 t2 Hstrict Hpassed.
  unfold validate_distribution in *.
  simpl in *.
  (* If |ratio - 0.5| <= t1 and t1 < t2, then |ratio - 0.5| <= t2 *)
  destruct (Nat.eqb (length (bytes_to_bits bytes)) 0) eqn:Heq.
  - (* Empty case *)
    simpl. destruct (Rle_dec _ _); reflexivity.
  - (* Non-empty *)
    destruct (Rle_dec (Rabs _) t1) eqn:Hdec1;
    destruct (Rle_dec (Rabs _) t2) eqn:Hdec2;
    try reflexivity.
    + (* t1 passed, but t2 failed — contradiction *)
      exfalso.
      apply Rle_dec_false in Hdec2.
      apply Rle_dec_true in Hdec1.
      lra.
Qed.

(* T7: Perfect balance (50% ones) always passes *)
Theorem perfect_balance_passes : forall bits,
  length bits > 0 ->
  2 * count_ones bits = length bits ->
  passed (validate_distribution
    (* Convert bits back to bytes - requires helper *)
    [] (* placeholder *)
    TOLERANCE) = true.
Proof.
  intros bits Hlen Hbalance.
  (* ones_ratio = 0.5 → |0.5 - 0.5| = 0 <= TOLERANCE *)
  admit.
Admitted.

(* T8: Soundness - validation passing implies statistical randomness *)
(* This requires probabilistic reasoning - axiomatized *)
Axiom validation_soundness : forall bytes source,
  source = QuantumVacuum ->
  passed (validate_distribution bytes TOLERANCE) = true ->
  (* Probabilistic statement: source is NISQ-grade random *)
  True. (* Placeholder for full probability theory *)

(* T9: Completeness - true random source passes with high probability *)
(* Requires Chernoff bounds / concentration inequalities *)
Axiom validation_completeness : forall bytes source,
  source = QuantumVacuum ->
  length bytes >= 32 -> (* Minimum batch size *)
  (* With probability ≥ 0.999, validation passes *)
  True. (* Placeholder for full probability theory *)

(* ═══════════════════════════════════════════════════════════════════════════
   HELPER LEMMAS
   ═══════════════════════════════════════════════════════════════════════════ *)

(* Bit count monotonicity *)
Lemma count_ones_app : forall l1 l2,
  count_ones (l1 ++ l2) = count_ones l1 + count_ones l2.
Proof.
  induction l1; intros l2.
  - simpl. reflexivity.
  - simpl. destruct a; simpl; rewrite IHl1; lia.
Qed.

(* Length of flattened bit list *)
Lemma bytes_to_bits_length : forall bytes,
  length (bytes_to_bits bytes) = 8 * length bytes.
Proof.
  induction bytes.
  - simpl. reflexivity.
  - simpl. unfold bytes_to_bits in *.
    rewrite flat_map_concat_map.
    rewrite app_length.
    (* Use byte_to_8bits_length *)
    admit.
Admitted.

End EntropyValidation.