Library Coq.micromega.QMicromega

Require Import OrderedRing.
Require Import RingMicromega.
Require Import Refl.
Require Import QArith.
Require Import Qfield.

Lemma Qsor : SOR 0 1 Qplus Qmult Qminus Qopp Qeq Qle Qlt.

Lemma QSORaddon :
  SORaddon 0 1 Qplus Qmult Qminus Qopp Qeq Qle
  0 1 Qplus Qmult Qminus Qopp
  Qeq_bool Qle_bool
  (fun x => x) (fun x => x) (pow_N 1 Qmult).

Require Import EnvRing.

Fixpoint Qeval_expr (env: PolEnv Q) (e: PExpr Q) : Q :=
  match e with
    | PEc c => c
    | PEX _ j => env j
    | PEadd pe1 pe2 => (Qeval_expr env pe1) + (Qeval_expr env pe2)
    | PEsub pe1 pe2 => (Qeval_expr env pe1) - (Qeval_expr env pe2)
    | PEmul pe1 pe2 => (Qeval_expr env pe1) * (Qeval_expr env pe2)
    | PEopp pe1 => - (Qeval_expr env pe1)
    | PEpow pe1 n => Qpower (Qeval_expr env pe1) (Z.of_N n)
  end.

Lemma Qeval_expr_simpl : forall env e,
  Qeval_expr env e =
  match e with
    | PEc c => c
    | PEX _ j => env j
    | PEadd pe1 pe2 => (Qeval_expr env pe1) + (Qeval_expr env pe2)
    | PEsub pe1 pe2 => (Qeval_expr env pe1) - (Qeval_expr env pe2)
    | PEmul pe1 pe2 => (Qeval_expr env pe1) * (Qeval_expr env pe2)
    | PEopp pe1 => - (Qeval_expr env pe1)
    | PEpow pe1 n => Qpower (Qeval_expr env pe1) (Z.of_N n)
  end.

Definition Qeval_expr' := eval_pexpr Qplus Qmult Qminus Qopp (fun x => x) (fun x => x) (pow_N 1 Qmult).

Lemma QNpower : forall r n, r ^ Z.of_N n = pow_N 1 Qmult r n.

Lemma Qeval_expr_compat : forall env e, Qeval_expr env e = Qeval_expr' env e.

Definition Qeval_op2 (o : Op2) : Q -> Q -> Prop :=
match o with
| OpEq => Qeq
| OpNEq => fun x y => ~ x == y
| OpLe => Qle
| OpGe => fun x y => Qle y x
| OpLt => Qlt
| OpGt => fun x y => Qlt y x
end.

Definition Qeval_formula (e:PolEnv Q) (ff : Formula Q) :=
  let (lhs,o,rhs) := ff in Qeval_op2 o (Qeval_expr e lhs) (Qeval_expr e rhs).

Definition Qeval_formula' :=
  eval_formula Qplus Qmult Qminus Qopp Qeq Qle Qlt (fun x => x) (fun x => x) (pow_N 1 Qmult).

Lemma Qeval_formula_compat : forall env f, Qeval_formula env f <-> Qeval_formula' env f.

Definition Qeval_nformula :=
  eval_nformula 0 Qplus Qmult Qeq Qle Qlt (fun x => x) .

Definition Qeval_op1 (o : Op1) : Q -> Prop :=
match o with
| Equal => fun x : Q => x == 0
| NonEqual => fun x : Q => ~ x == 0
| Strict => fun x : Q => 0 < x
| NonStrict => fun x : Q => 0 <= x
end.

Lemma Qeval_nformula_dec : forall env d, (Qeval_nformula env d) \/ ~ (Qeval_nformula env d).

Definition QWitness := Psatz Q.

Definition QWeakChecker := check_normalised_formulas 0 1 Qplus Qmult Qeq_bool Qle_bool.

Require Import List.

Lemma QWeakChecker_sound : forall (l : list (NFormula Q)) (cm : QWitness),
  QWeakChecker l cm = true ->
  forall env, make_impl (Qeval_nformula env) l False.

Require Import Coq.micromega.Tauto.

Definition Qnormalise := @cnf_normalise Q 0 1 Qplus Qmult Qminus Qopp Qeq_bool.
Definition Qnegate := @cnf_negate Q 0 1 Qplus Qmult Qminus Qopp Qeq_bool.

Definition qunsat := check_inconsistent 0 Qeq_bool Qle_bool.

Definition qdeduce := nformula_plus_nformula 0 Qplus Qeq_bool.

Definition QTautoChecker (f : BFormula (Formula Q)) (w: list QWitness) : bool :=
  @tauto_checker (Formula Q) (NFormula Q)
  qunsat qdeduce
  Qnormalise
  Qnegate QWitness QWeakChecker f w.

Lemma QTautoChecker_sound : forall f w, QTautoChecker f w = true -> forall env, eval_f (Qeval_formula env) f.