# Properties of the primitive operators for the Binary64 format

Notation valid_binary := (valid_binary prec emax).

Definition SF64classify := SFclassify prec.
Definition SF64mul := SFmul prec emax.
Definition SF64sub := SFsub prec emax.
Definition SF64div := SFdiv prec emax.
Definition SF64sqrt := SFsqrt prec emax.
Definition SF64succ := SFsucc prec emax.
Definition SF64pred := SFpred prec emax.

Axiom Prim2SF_valid : forall x, valid_binary (Prim2SF x) = true.
Axiom SF2Prim_Prim2SF : forall x, SF2Prim (Prim2SF x) = x.
Axiom Prim2SF_SF2Prim : forall x, valid_binary x = true -> Prim2SF (SF2Prim x) = x.

Theorem Prim2SF_inj : forall x y, Prim2SF x = Prim2SF y -> x = y.

Theorem SF2Prim_inj : forall x y, SF2Prim x = SF2Prim y -> valid_binary x = true -> valid_binary y = true -> x = y.

Axiom opp_spec : forall x, Prim2SF (-x)%float = SFopp (Prim2SF x).
Axiom abs_spec : forall x, Prim2SF (abs x) = SFabs (Prim2SF x).

Axiom eqb_spec : forall x y, (x == y)%float = SFeqb (Prim2SF x) (Prim2SF y).
Axiom ltb_spec : forall x y, (x < y)%float = SFltb (Prim2SF x) (Prim2SF y).
Axiom leb_spec : forall x y, (x <= y)%float = SFleb (Prim2SF x) (Prim2SF y).

Definition flatten_cmp_opt c :=
match c with
| None => FNotComparable
| Some Eq => FEq
| Some Lt => FLt
| Some Gt => FGt
end.
Axiom compare_spec : forall x y, (x ?= y)%float = flatten_cmp_opt (SFcompare (Prim2SF x) (Prim2SF y)).

Axiom classify_spec : forall x, classify x = SF64classify (Prim2SF x).
Axiom mul_spec : forall x y, Prim2SF (x * y)%float = SF64mul (Prim2SF x) (Prim2SF y).
Axiom add_spec : forall x y, Prim2SF (x + y)%float = SF64add (Prim2SF x) (Prim2SF y).
Axiom sub_spec : forall x y, Prim2SF (x - y)%float = SF64sub (Prim2SF x) (Prim2SF y).
Axiom div_spec : forall x y, Prim2SF (x / y)%float = SF64div (Prim2SF x) (Prim2SF y).
Axiom sqrt_spec : forall x, Prim2SF (sqrt x) = SF64sqrt (Prim2SF x).

Axiom of_int63_spec : forall n, Prim2SF (of_int63 n) = binary_normalize prec emax (to_Z n) 0%Z false.
Axiom normfr_mantissa_spec : forall f, to_Z (normfr_mantissa f) = Z.of_N (SFnormfr_mantissa prec (Prim2SF f)).

Axiom frshiftexp_spec : forall f, let (m,e) := frshiftexp f in (Prim2SF m, ((to_Z e) - shift)%Z) = SFfrexp prec emax (Prim2SF f).
Axiom ldshiftexp_spec : forall f e, Prim2SF (ldshiftexp f e) = SFldexp prec emax (Prim2SF f) ((to_Z e) - shift).

Axiom next_up_spec : forall x, Prim2SF (next_up x) = SF64succ (Prim2SF x).
Axiom next_down_spec : forall x, Prim2SF (next_down x) = SF64pred (Prim2SF x).