Section Int31_Specs.
Local Open Scope Z_scope.
Notation "[| x |]" := (phi x)  (at level 0, x at level 99).
Local Notation wB := (2 ^ (Z.of_nat size)).
Notation "[+| c |]" :=
(interp_carry 1 wB phi c)  (at level 0, c at level 99).
Notation "[-| c |]" :=
(interp_carry (-1) wB phi c)  (at level 0, c at level 99).
Notation "[|| x ||]" :=
(zn2z_to_Z wB phi x)  (at level 0, x at level 99).
(** Subtraction *)
(** Multiplication *)
(** Division *)
Fixpoint addmuldiv31_alt n i j :=
match n with
| O => i
| S n => addmuldiv31_alt n (sneakl (firstl j) i) (shiftl j)
end.

Lemma spec_add_mul_div : forall x y p, [|p|] <= Zpos 31 ->
[| addmuldiv31 p x y |] =
([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ((Zpos 31) - [|p|]))) mod wB.