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(*** Barrett Reduction *)
(** This file implements Barrett Reduction on [ZLikeOps]. We follow
[BarretReduction/ZHandbook.v]. *)
Require Import Coq.ZArith.ZArith Coq.Lists.List Coq.Classes.Morphisms Coq.micromega.Psatz.
Require Import Crypto.ModularArithmetic.BarrettReduction.ZHandbook.
Require Import Crypto.ModularArithmetic.Pow2Base.
Require Import Crypto.ModularArithmetic.ExtendedBaseVector.
Require Import Crypto.ModularArithmetic.ZBounded.
Require Import Crypto.BaseSystem.
Require Import Crypto.Util.ZUtil.
Require Import Crypto.Util.Tactics.
Require Import Crypto.Util.Notations.
Local Open Scope small_zlike_scope.
Local Open Scope large_zlike_scope.
Local Open Scope Z_scope.
Section barrett.
Context (m b k μ offset : Z)
(m_pos : 0 < m)
(base_pos : 0 < b)
(k_good : m < b^k)
(μ_good : μ = b^(2*k) / m) (* [/] is [Z.div], which is truncated *)
(offset_nonneg : 0 <= offset)
(k_big_enough : offset <= k)
(m_small : 3 * m <= b^(k+offset))
(m_large : b^(k-offset) <= m + 1).
Context {ops : ZLikeOps (b^(k+offset)) (b^(k-offset)) m} {props : ZLikeProperties ops}
(μ' : SmallT)
(μ'_good : small_valid μ')
(μ'_eq : decode_small μ' = μ).
Definition barrett_reduce : forall x : LargeT,
{ barrett_reduce : SmallT
| medium_valid x
-> decode_small barrett_reduce = (decode_large x) mod m
/\ small_valid barrett_reduce }.
Proof.
intro x. evar (pr : SmallT); exists pr. intros x_valid.
assert (0 <= decode_large x < b^(k+offset) * b^(k-offset)) by auto using decode_medium_valid.
assert (0 <= decode_large x < b^(2 * k)) by (autorewrite with pull_Zpow zsimplify in *; omega).
assert ((decode_large x) mod b^(k-offset) < b^(k-offset)) by auto with zarith omega.
rewrite (barrett_reduction_small m b (decode_large x) k μ offset) by omega.
rewrite <- μ'_eq.
pull_zlike_decode; cbv zeta; pull_zlike_decode. (* Extra [cbv iota; pull_zlike_decode] to work around bug #4165 (https://coq.inria.fr/bugs/show_bug.cgi?id=4165) in 8.4 *)
subst pr; split; [ reflexivity | exact _ ].
Defined.
End barrett.
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