Library Coq.micromega.RMicromega

Require Import OrderedRing.
Require Import RingMicromega.
Require Import Refl.
Require Import Raxioms RIneq Rpow_def DiscrR.
Require Import QArith.
Require Import Qfield.
Require Import Qreals.

Require Setoid.

Definition Rsrt : ring_theory R0 R1 Rplus Rmult Rminus Ropp (@eq R).

Open Scope R_scope.

Lemma Rsor : SOR R0 R1 Rplus Rmult Rminus Ropp (@eq R) Rle Rlt.

Notation IQR := Q2R (only parsing).

Lemma Rinv_1 : forall x, x * / 1 = x.

Lemma Qeq_true : forall x y, Qeq_bool x y = true -> IQR x = IQR y.

Lemma Qeq_false : forall x y, Qeq_bool x y = false -> IQR x <> IQR y.

Lemma Qle_true : forall x y : Q, Qle_bool x y = true -> IQR x <= IQR y.

Lemma IQR_0 : IQR 0 = 0.

Lemma IQR_1 : IQR 1 = 1.

Lemma IQR_inv_ext : forall x,
  IQR (/ x) = (if Qeq_bool x 0 then 0 else / IQR x).

Notation to_nat := N.to_nat.

Lemma QSORaddon :
  @SORaddon R
  R0 R1 Rplus Rmult Rminus Ropp (@eq R) Rle
  Q 0%Q 1%Q Qplus Qmult Qminus Qopp
  Qeq_bool Qle_bool
  IQR nat to_nat pow.

Inductive Rcst :=
| C0
| C1
| CQ (r : Q)
| CZ (r : Z)
| CPlus (r1 r2 : Rcst)
| CMinus (r1 r2 : Rcst)
| CMult (r1 r2 : Rcst)
| CInv (r : Rcst)
| COpp (r : Rcst).

Fixpoint Q_of_Rcst (r : Rcst) : Q :=
  match r with
    | C0 => 0 # 1
    | C1 => 1 # 1
    | CZ z => z # 1
    | CQ q => q
    | CPlus r1 r2 => Qplus (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CMinus r1 r2 => Qminus (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CMult r1 r2 => Qmult (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CInv r => Qinv (Q_of_Rcst r)
    | COpp r => Qopp (Q_of_Rcst r)
  end.

Fixpoint R_of_Rcst (r : Rcst) : R :=
  match r with
    | C0 => R0
    | C1 => R1
    | CZ z => IZR z
    | CQ q => IQR q
    | CPlus r1 r2 => (R_of_Rcst r1) + (R_of_Rcst r2)
    | CMinus r1 r2 => (R_of_Rcst r1) - (R_of_Rcst r2)
    | CMult r1 r2 => (R_of_Rcst r1) * (R_of_Rcst r2)
    | CInv r =>
      if Qeq_bool (Q_of_Rcst r) (0 # 1)
        then R0
        else Rinv (R_of_Rcst r)
      | COpp r => - (R_of_Rcst r)
  end.

Lemma Q_of_RcstR : forall c, IQR (Q_of_Rcst c) = R_of_Rcst c.

Require Import EnvRing.

Definition INZ (n:N) : R :=
  match n with
    | N0 => IZR 0%Z
    | Npos p => IZR (Zpos p)
  end.

Definition Reval_expr := eval_pexpr Rplus Rmult Rminus Ropp R_of_Rcst N.to_nat pow.

Definition Reval_op2 (o:Op2) : R -> R -> Prop :=
    match o with
      | OpEq => @eq R
      | OpNEq => fun x y => ~ x = y
      | OpLe => Rle
      | OpGe => Rge
      | OpLt => Rlt
      | OpGt => Rgt
    end.

Definition Reval_formula (e: PolEnv R) (ff : Formula Rcst) :=
  let (lhs,o,rhs) := ff in Reval_op2 o (Reval_expr e lhs) (Reval_expr e rhs).

Definition Reval_formula' :=
  eval_sformula Rplus Rmult Rminus Ropp (@eq R) Rle Rlt N.to_nat pow R_of_Rcst.

Definition QReval_formula :=
  eval_formula Rplus Rmult Rminus Ropp (@eq R) Rle Rlt IQR N.to_nat pow .

Lemma Reval_formula_compat : forall env f, Reval_formula env f <-> Reval_formula' env f.

Definition Qeval_nformula :=
  eval_nformula 0 Rplus Rmult (@eq R) Rle Rlt IQR.

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

Definition RWitness := Psatz Q.

Definition RWeakChecker := check_normalised_formulas 0%Q 1%Q Qplus Qmult Qeq_bool Qle_bool.

Require Import List.

Lemma RWeakChecker_sound : forall (l : list (NFormula Q)) (cm : RWitness),
  RWeakChecker l cm = true ->
  forall env, make_impl (Qeval_nformula env) l False.

Require Import Coq.micromega.Tauto.

Definition Rnormalise := @cnf_normalise Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq_bool.
Definition Rnegate := @cnf_negate Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq_bool.

Definition runsat := check_inconsistent 0%Q Qeq_bool Qle_bool.

Definition rdeduce := nformula_plus_nformula 0%Q Qplus Qeq_bool.

Definition RTautoChecker (f : BFormula (Formula Rcst)) (w: list RWitness) : bool :=
  @tauto_checker (Formula Q) (NFormula Q)
  runsat rdeduce
  Rnormalise Rnegate
  RWitness RWeakChecker (map_bformula (map_Formula Q_of_Rcst) f) w.

Lemma RTautoChecker_sound : forall f w, RTautoChecker f w = true -> forall env, eval_f (Reval_formula env) f.