(* Begin Micromega *)
Inductive Op1 : Set := (* relations with 0 *)
| Equal (* == 0 *)
| NonEqual (* ~= 0 *)
| Strict (* > 0 *)
| NonStrict (* >= 0 *).
Definition NFormula := (PolC * Op1)%type. (* normalized formula *)
Definition eval_op1 (o : Op1) : R -> Prop :=
match o with
| Equal => fun x => x == 0
| NonEqual => fun x : R => x ~= 0
| Strict => fun x : R => 0 < x
| NonStrict => fun x : R => 0 <= x
end.
Definition eval_nformula (env : PolEnv) (f : NFormula) : Prop :=
let (p, op) := f in eval_op1 op (eval_pol env p).
(** Rule of "signs" for addition and multiplication. An arbitrary result is coded buy None. *)
Definition OpMult (o o' : Op1) : option Op1 :=
match o with
| Equal => Some Equal
| NonStrict =>
match o' with
| Equal => Some Equal
| NonEqual  => None
| Strict    => Some NonStrict
| NonStrict => Some NonStrict
end
| Strict => match o' with
| NonEqual => None
|  _       => Some o'
end
| NonEqual => match o' with
| Equal => Some Equal
| NonEqual => Some NonEqual
| _        => None
end
end.
Definition OpAdd (o o': Op1) : option Op1 :=
match o with
| Equal => Some o'
| NonStrict =>
match o' with
| Strict => Some Strict
| NonEqual => None
| _ => Some NonStrict
end
| Strict => match o' with
| NonEqual => None
|  _        => Some Strict
end
| NonEqual => match o' with
| Equal  => Some NonEqual
| _      => None
end
end.

(* Specifications: *)
Lemma OpMult_sound :
forall (o o' om: Op1) (x y : R),
eval_op1 o x -> eval_op1 o' y -> OpMult o o' = Some om -> eval_op1 om (x * y).
Lemma OpAdd_sound :
forall (o o' oa : Op1) (e e' : R),
eval_op1 o e -> eval_op1 o' e' -> OpAdd o o' = Some oa -> eval_op1 oa (e + e').