The Coq Proof Assistant

# 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_pop2 (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 Qlt_bool (x y : Q) :=
(Qnum x * QDen y <? Qnum y * QDen x)%Z.
Definition Qeval_bop2 (o : Op2) : Q -> Q -> bool :=
match o with
| OpEq => Qeq_bool
| OpNEq => fun x y => negb (Qeq_bool x y)
| OpLe => Qle_bool
| OpGe => fun x y => Qle_bool y x
| OpLt => Qlt_bool
| OpGt => fun x y => Qlt_bool y x
end.
Lemma Qlt_bool_iff : forall q1 q2,
Qlt_bool q1 q2 = true <-> q1 < q2.
Lemma pop2_bop2 :
forall (op : Op2) (q1 q2 : Q), is_true (Qeval_bop2 op q1 q2) <-> Qeval_pop2 op q1 q2.
Definition Qeval_op2 (k:Tauto.kind) : Op2 -> Q -> Q -> Tauto.rtyp k:=
if k as k0 return (Op2 -> Q -> Q -> Tauto.rtyp k0)
then Qeval_pop2 else Qeval_bop2.
Lemma Qeval_op2_hold : forall k op q1 q2,
Tauto.hold k (Qeval_op2 k op q1 q2) <-> Qeval_pop2 op q1 q2.
Definition Qeval_formula (e:PolEnv Q) (k: Tauto.kind) (ff : Formula Q) :=
let (lhs,o,rhs) := ff in Qeval_op2 k 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 b f, Tauto.hold b (Qeval_formula env b 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.
Register QWitness as micromega.QWitness.type.
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 Qle_bool.
Definition Qnegate := @cnf_negate Q 0 1 Qplus Qmult Qminus Qopp Qeq_bool Qle_bool.
Definition qunsat := check_inconsistent 0 Qeq_bool Qle_bool.
Definition qdeduce := nformula_plus_nformula 0 Qplus Qeq_bool.
Definition normQ := norm 0 1 Qplus Qmult Qminus Qopp Qeq_bool.
Definition cnfQ (Annot:Type) (TX: Tauto.kind -> Type) (AF: Type) (k: Tauto.kind) (f: TFormula (Formula Q) Annot TX AF k) :=
rxcnf qunsat qdeduce (Qnormalise Annot) (Qnegate Annot) true f.
Definition QTautoChecker (f : BFormula (Formula Q) Tauto.isProp) (w: list QWitness) : bool :=
@tauto_checker (Formula Q) (NFormula Q) unit
qunsat qdeduce
(Qnormalise unit)
(Qnegate unit) QWitness (fun cl => QWeakChecker (List.map fst cl)) f w.
Lemma QTautoChecker_sound : forall f w, QTautoChecker f w = true -> forall env, eval_bf (Qeval_formula env) f.