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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
(*                                                                      *)
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(*Declare ML Module "micromega_plugin".*)
(*Definition Zeval_expr :=  eval_pexpr 0 Z.add Z.mul Z.sub Z.opp  (fun x => x) (fun x => Z.of_N x) (Z.pow).*)
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.
Definition Qeval_expr' := eval_pexpr  Qplus Qmult Qminus Qopp (fun x => x) (fun x => x) (pow_N 1 Qmult).
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).
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.
Definition QWitness := Psatz Q.
Definition QWeakChecker := check_normalised_formulas 0 1 Qplus Qmult Qeq_bool Qle_bool.

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