(* --------------------------------------------------------------------
 * Copyright IMDEA Software Institute / INRIA - 2013, 2014
 * -------------------------------------------------------------------- *)

require Int.
require import FSet.

type word.

const length:int.
axiom leq0_length: Int.(<=) 0 length.

op zeros: word.
op ones: word.

op ( ^ ): word -> word -> word.
op land : word -> word -> word.
op lopp (x:word) = x.

axiom ones_neq0 : ones <> zeros.

axiom xorwA (w1 w2 w3:word):
  w1 ^ (w2 ^ w3) = (w1 ^ w2) ^ w3.
axiom xorwC (w1 w2:word):
  w1 ^ w2 = w2 ^ w1.
axiom xor0w x: zeros ^ x = x.
axiom xorwK (w:word):
  w ^ w = zeros.

lemma nosmt xorw0 (w:word): w ^ zeros = w
by [].

lemma nosmt xorNw (x:word): (lopp x) ^ x = zeros
by [].

axiom landwA x y z: land x (land y z) = land (land x y) z.
axiom landwC x y: land x y = land y x.
axiom land1w x: land ones x = x.
axiom landwDl (x y z:word): land (x ^ y) z = land x z ^ land y z.
axiom landI (x:word): land x x = x.

lemma subwE : ( ^ ) = fun (x y: word), x ^ lopp y.
proof.
  rewrite -ExtEq.fun_ext => x; rewrite -ExtEq.fun_ext => y; smt.
qed.

(** View bitstring as a ring *)
require (*--*) Ring.
(*---*) import Ring.BoolRing.

instance bring with word
  op rzero = zeros
  op rone  = ones
  op add   = (   ^  )
  op mul   = land
  op opp   = lopp

  proof oner_neq0 by smt
  proof addr0     by smt
  proof addrA     by smt
  proof addrC     by smt
  proof addrK     by smt
  proof mulr1     by smt
  proof mulrA     by smt
  proof mulrC     by smt
  proof mulrDl    by smt
  proof mulrK     by smt
  proof oppr_id   by smt.

require export ABitstring.
op to_bits: word -> bitstring.
op from_bits: bitstring -> word.

axiom length_to_bits w:
  `|to_bits w| = length.
axiom can_from_to w:
  from_bits (to_bits w) = w.
axiom pcan_to_from (b:bitstring):
  `|b| = length =>
  to_bits (from_bits b) = b.

(** Conversion with int *)
import Int.
op to_int   : word -> int.
op from_int : int -> word.

axiom to_from w: from_int (to_int w) = w.
axiom from_to i: to_int (from_int i) = i %% 2^length.

lemma from_to_bound i:
   0 <= i < 2^length =>
   to_int (from_int i) = i.
proof -strict.
 rewrite from_to.
 intros H.
 elim (EuclDiv.ediv_unique i (2^length) 0 i _ _ _) => //;first 2 smt.
 by intros _ <-.
qed.

(** univ *)

theory Univ.
  op univ = FSet.img from_int (Interval.interval 0  (2^length -1)).

  lemma mem_univ w: mem w univ.
  proof.
    rewrite /univ img_def; exists (to_int w);rewrite to_from Interval.mem_interval/=.
    by rewrite -to_from from_to;smt.
  qed.

  require import ISet.

  lemma finite_univ : Finite.finite (ISet.univ <:word>).
  proof. by exists Univ.univ => x;rewrite ISet.mem_univ Univ.mem_univ. qed.

end Univ.
 
theory Dword.
  require import Distr.
  require import Real.

  op dword: word distr.
  axiom mu_x_def w: mu_x dword w = 1%r/(2^length)%r.
  axiom lossless: weight dword = 1%r.

  lemma in_supp_def w: in_supp w dword.
  proof -strict.
  rewrite /in_supp mu_x_def;smt.
  qed.

  lemma mu_cpMemw X:
    mu dword (cpMem X) = (card X)%r / (2^length)%r.
  proof -strict.
  by rewrite (mu_cpMem _ _ (1%r/(2^length)%r))=> // x;
     rewrite mu_x_def.
  qed.

  import FSet.Dexcepted.
  lemma lossless_restrw X:
    card X < 2^length =>
    weight (dword \ X) = 1%r.
  proof -strict.
  intros=> card_X; rewrite lossless_restr ?lossless // ?mu_cpMemw;
  cut <-: (forall x y, x * (1%r / y) = x / y) by smt;
  apply (real_lt_trans _ ((2^length)%r* (1%r/(2^length)%r)) _); last smt.
  apply mulrM; last by rewrite from_intM.
  smt.
  qed.
end Dword.