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+(*** Barrett Reduction *)
+(** This file implements Barrett Reduction on [Z].  We follow Wikipedia. *)
+Require Import Coq.ZArith.ZArith Coq.micromega.Psatz.
+Require Import Crypto.Util.ZUtil.
+Require Import Crypto.Util.Tactics.BreakMatch.
+
+Local Open Scope Z_scope.
+
+Section barrett.
+  Context (n a : Z)
+          (n_reasonable : n <> 0).
+  (** Quoting Wikipedia <https://en.wikipedia.org/wiki/Barrett_reduction>: *)
+  (** In modular arithmetic, Barrett reduction is a reduction
+      algorithm introduced in 1986 by P.D. Barrett. A naive way of
+      computing *)
+  (** [c = a mod n] *)
+  (** would be to use a fast division algorithm. Barrett reduction is
+      an algorithm designed to optimize this operation assuming [n] is
+      constant, and [a < n²], replacing divisions by
+      multiplications. *)
+
+  (** * General idea *)
+  Section general_idea.
+    (** Let [m = 1 / n] be the inverse of [n] as a floating point
+        number. Then *)
+    (** [a mod n = a - ⌊a m⌋ n] *)
+    (** where [⌊ x ⌋] denotes the floor function. The result is exact,
+        as long as [m] is computed with sufficient accuracy. *)
+
+    (* [/] is [Z.div], which means truncated division *)
+    Local Notation "⌊am⌋" := (a / n) (only parsing).
+
+    Theorem naive_barrett_reduction_correct
+      : a mod n = a - ⌊am⌋ * n.
+    Proof using n_reasonable.
+      apply Zmod_eq_full; assumption.
+    Qed.
+  End general_idea.
+
+  (** * Barrett algorithm *)
+  Section barrett_algorithm.
+    (** Barrett algorithm is a fixed-point analog which expresses
+        everything in terms of integers. Let [k] be the smallest
+        integer such that [2ᵏ > n]. Think of [n] as representing the
+        fixed-point number [n 2⁻ᵏ]. We precompute [m] such that [m =
+        ⌊4ᵏ / n⌋]. Then [m] represents the fixed-point number
+        [m 2⁻ᵏ ≈ (n 2⁻ᵏ)⁻¹]. *)
+    (** N.B. We don't need [k] to be the smallest such integer. *)
+    Context (k : Z)
+            (k_good : n < 2 ^ k)
+            (m : Z)
+            (m_good : m = 4^k / n). (* [/] is [Z.div], which is truncated *)
+    (** Wikipedia neglects to mention non-negativity, but we need it.
+        It might be possible to do with a relaxed assumption, such as
+        the sign of [a] and the sign of [n] being the same; but I
+        figured it wasn't worth it. *)
+    Context (n_pos : 0 < n) (* or just [0 <= n], since we have [n <> 0] above *)
+            (a_nonneg : 0 <= a).
+
+    Lemma k_nonnegative : 0 <= k.
+    Proof using Type*.
+      destruct (Z_lt_le_dec k 0); try assumption.
+      rewrite !Z.pow_neg_r in * by lia; lia.
+    Qed.
+
+    (** Now *)
+    Let q := (m * a) / 4^k.
+    Let r := a - q * n.
+    (** Because of the floor function (in Coq, because [/] means
+        truncated division), [q] is an integer and [r ≡ a mod n]. *)
+    Theorem barrett_reduction_equivalent
+      : r mod n = a mod n.
+    Proof using m_good.
+      subst r q m.
+      rewrite <- !Z.add_opp_r, !Zopp_mult_distr_l, !Z_mod_plus_full by assumption.
+      reflexivity.
+    Qed.
+
+    Lemma qn_small
+      : q * n <= a.
+    Proof using a_nonneg k_good m_good n_pos n_reasonable.
+      pose proof k_nonnegative; subst q r m.
+      assert (0 <= 2^(k-1)) by zero_bounds.
+      Z.simplify_fractions_le.
+    Qed.
+
+    (** Also, if [a < n²] then [r < 2n]. *)
+    (** N.B. It turns out that it is sufficient to assume [a < 4ᵏ]. *)
+    Context (a_small : a < 4^k).
+    Lemma q_nice : { b : bool | q = a / n + if b then -1 else 0 }.
+    Proof using a_nonneg a_small k_good m_good n_pos n_reasonable.
+      assert (0 <= (4 ^ k * a / n) mod 4 ^ k < 4 ^ k) by auto with zarith lia.
+      assert (0 <= a * (4 ^ k mod n) / n < 4 ^ k) by (auto with zero_bounds zarith lia).
+      subst q r m.
+      rewrite (Z.div_mul_diff_exact''' (4^k) n a) by lia.
+      rewrite (Z_div_mod_eq (4^k * _ / n) (4^k)) by lia.
+      autorewrite with push_Zmul push_Zopp zsimplify zstrip_div.
+      eexists; reflexivity.
+    Qed.
+
+    Lemma r_small : r < 2 * n.
+    Proof using a_nonneg a_small k_good m_good n_pos n_reasonable q.
+      Hint Rewrite (Z.mul_div_eq' a n) using lia : zstrip_div.
+      assert (a mod n < n) by auto with zarith lia.
+      unfold r; rewrite (proj2_sig q_nice); generalize (proj1_sig q_nice); intro; subst q m.
+      autorewrite with push_Zmul zsimplify zstrip_div.
+      break_match; auto with lia.
+    Qed.
+
+    (** In that case, we have *)
+    Theorem barrett_reduction_small
+      : a mod n = if r <? n
+                  then r
+                  else r - n.
+    Proof using a_nonneg a_small k_good m_good n_pos n_reasonable q.
+      pose proof r_small. pose proof qn_small.
+      destruct (r <? n) eqn:rlt; Z.ltb_to_lt.
+      { symmetry; apply (Zmod_unique a n q); subst r; lia. }
+      { symmetry; apply (Zmod_unique a n (q + 1)); subst r; lia. }
+    Qed.
+  End barrett_algorithm.
+End barrett.