Section Micromega.
(* Assume we have a strict(ly?) ordered ring *)
Variable R : Type.
Variables rO rI : R.
Variables rplus rtimes rminus: R -> R -> R.
Variable ropp : R -> R.
Variables req rle rlt : R -> R -> Prop.
Variable sor : SOR rO rI rplus rtimes rminus ropp req rle rlt.
Notation "0" := rO.
Notation "1" := rI.
Notation "x + y" := (rplus x y).
Notation "x * y " := (rtimes x y).
Notation "x - y " := (rminus x y).
Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
(* Assume we have a type of coefficients C and a morphism from C to R *)
Variable C : Type.
Variables cO cI : C.
Variables cplus ctimes cminus: C -> C -> C.
Variable copp : C -> C.
Variables ceqb cleb : C -> C -> bool.
Variable phi : C -> R.
(* Power coefficients *)
Variable E : Type. (* the type of exponents *)
Variable pow_phi : N -> E.
Variable rpow : R -> E -> R.
Notation "[ x ]" := (phi x).
Notation "x [=] y" := (ceqb x y).
Notation "x [<=] y" := (cleb x y).
(* Let's collect all hypotheses in addition to the ordered ring axioms into one structure *)
Record SORaddon := mk_SOR_addon {
SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
SORpower : power_theory rI rtimes req pow_phi rpow;
SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.
Variable addon : SORaddon.
Add Relation R req
reflexivity proved by (@Equivalence_Reflexive _ _ (SORsetoid sor))
symmetry proved by (@Equivalence_Symmetric _ _ (SORsetoid sor))
transitivity proved by (@Equivalence_Transitive _ _ (SORsetoid sor))
as micomega_sor_setoid.
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.
Add Morphism ropp with signature req ==> req as ropp_morph.
Add Morphism rle with signature req ==> req ==> iff as rle_morph.
Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.
Add Morphism rminus with signature req ==> req ==> req as rminus_morph.
Definition cneqb (x y : C) := negb (ceqb x y).
Definition cltb (x y : C) := (cleb x y) && (cneqb x y).
Notation "x [~=] y" := (cneqb x y).
Notation "x [<] y" := (cltb x y).
Ltac le_less := rewrite (Rle_lt_eq sor); left; try assumption.
Ltac le_equal := rewrite (Rle_lt_eq sor); right; try reflexivity; try assumption.
Ltac le_elim H := rewrite (Rle_lt_eq sor) in H; destruct H as [H | H].
(* Begin Micromega *)
Definition PolC := Pol C. (* polynomials in generalized Horner form, defined in Ring_polynom or EnvRing *)
Definition PolEnv := Env R. (* For interpreting PolC *)
Definition eval_pol : PolEnv -> PolC -> R :=
Pphi rplus rtimes phi.
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.

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').