Require Import Coq.Lists.List.
Require Import Coq.ZArith.ZArith Coq.ZArith.Zdiv.
Require Import Coq.omega.Omega Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Arith.
Require Import Crypto.Util.ListUtil.
Require Import Crypto.Util.Notations.
Require Export Crypto.Util.FixCoqMistakes.
Import Nat.

Local Open Scope Z.

Class BaseVector (base : list Z):= {
  base_positive : forall b, In b base -> b > 0; (* nonzero would probably work too... *)
  b0_1 : forall x, nth_default x base 0 = 1; (** TODO(jadep,jgross): change to [nth_error base 0 = Some 1], then use [nth_error_value_eq_nth_default] to prove a [forall x, nth_default x base 0 = 1] as a lemma *)
  base_good :
    forall i j, (i+j < length base)%nat ->
    let b := nth_default 0 base in
    let r := (b i * b j) / b (i+j)%nat in
    b i * b j = r * b (i+j)%nat
}.

Section BaseSystem.
  Context (base : list Z).
  (** [BaseSystem] implements an constrained positional number system.
      A wide variety of bases are supported: the base coefficients are not
      required to be powers of 2, and it is NOT necessarily the case that
      $b_{i+j} = b_i b_j$. Implementations of addition and multiplication are
      provided, with focus on near-optimal multiplication performance on
      non-trivial but small operands: maybe 10 32-bit integers or so. This
      module does not handle carries automatically: if no restrictions are put
      on the use of a [BaseSystem], each digit is unbounded. This has nothing
      to do with modular arithmetic either.
  *)
  Definition digits : Type := list Z.

  Definition accumulate p acc := fst p * snd p + acc.
  Definition decode' bs u  := fold_right accumulate 0 (combine u bs).
  Definition decode := decode' base.
  Definition mul_each u := map (Z.mul u).
End BaseSystem.